Systems of Equations

I. Intro to Systems of Equations

A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.

In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.

2x + y = 15

3x - y = 5

The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.

2(4) + (7) = 15 True

3(4) - (7) = 5 True

In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. 

II. Types of Linear Systems

A consistent system of equations has at least one solution. 

There are two types of consistent system of equations. 

• A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. 

• A consistent system is considered to be a dependent system if the equations have the same slope and the same y-intercepts. In other words, they are the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.

Another type of system of linear equations is an inconsistent system, which is one in which the equations represent two parallel lines. The lines have the same slope and different y-intercepts. There are no points common to both lines; hence, there is no solution to the system.

III. Determining Whether an Ordered Pair is a Solutions to a System of Equations

In order for an ordered pair to a solution, it needs to be true for both equations. To test, you:

1. Substitute the ordered pair into each equation in the system.

2. Determine whether true statements result from the substitution in both equations; if so, the ordered pair is a solution.

Example

Determine whether the ordered pair (5,1) is a solution to the given system of equations.

𝑥 + 3⁢𝑦 = 8

2⁢𝑥 − 9 = 𝑦

Substitute the ordered pair (5, 1) into both equations.

(5) + 3⁢(1) = 8

8 = 8      True

2⁢(5) − 9 = (1)

1 = 1      True

The ordered pair (5,1) satisfies both equations, so it is the solution to the system.

We can see the solution clearly by plotting the graph of each equation. Since the solution is an ordered pair that satisfies both equations, it is a point on both of the lines and thus the point of intersection of the two lines.

IV. Solving Systems of Equations by Graphing

There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.

Example

Solve the following system of equations by graphing. Identify the type of system.

2⁢𝑥 + 𝑦 = −8
𝑥 − 𝑦 = −1

Solution:

Solve the first equation for 𝑦.

2⁢𝑥 + 𝑦 = −8

𝑦 = −2⁢𝑥 ⁢−8

Solve the second equation for 𝑦.
𝑥 − 𝑦 = −1

𝑦 = 𝑥 + 1

Graph both equations 

The lines appear to intersect at the point (−3,⁢−2). We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations.

2⁢(−3 ) + (−2) = −8 

−8 = −8               True

(−3) − (−2) = −1

−1 = −1               True

The solution to the system is the ordered pair (−3,⁢−2), so the system is independent.


V. Solving Systems of Equations by Substitution
Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will now look at the substitution method, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable.

Algorithm for solving a system of two equations in two variables using substitution

1. Solve one of the two equations for one of the variables in terms of the other.
2. Substitute the expression for this variable into the second equation, then solve for the remaining variable.
3. Substitute that solution into either of the original equations to find the value of the first variable. If possible, write the solution as an ordered pair.
4. Check the solution in both equations.

Example
Solve the following system of equations by substitution.

−𝑥 + 𝑦 = −5

2⁢𝑥 − 5⁢𝑦 = 1

First, we will solve the first equation for 𝑦.
−𝑥 + 𝑦 = −5

𝑦 = 𝑥 ⁢− 5

Now we can substitute the expression 𝑥⁢−5 for 𝑦 in the second equation.

2⁢𝑥 − 5⁢𝑦 = 1

2⁢𝑥 − 5⁢(𝑥 − 5) = 1

2⁢𝑥 − 5⁢𝑥 + 25 = 1

−3⁢𝑥 = −24

𝑥 = 8

Now, we substitute 𝑥 = 8 into the first equation and solve for 𝑦.
−(8) + 𝑦 = −5

𝑦 = 3

Our solution is (8,3).

Check the solution by substituting (8,3) into both equations.
−𝑥 + 𝑦 = −5

−(8) + (3) = −5     True

2⁢𝑥 − 5⁢𝑦 = 1

2⁢(8) − 5⁢(3) = 1     True


VI. Misc. 

A. Other Methods of Solving Systems of Equations

There are other methods of solving systems of equations which are taught in a full algebra class including the addition (elimination) method and matrix methods. 

B. System of Equations Word Problems

Khan: Systems of Equations Chores Word Problem

Lumen: Application of 2x2 Systems of Equations

Reference

OpenStax: System of Linear Equations

Khan: Systems of Equations

Lumen: Elementary Algebra

Functions

 

I. Functions

A relation is any set of ordered pairs, (x,y). The collection of all possible x-values in the ordered pairs is called the domain. The set of all possible y-values in the ordered pairs is called the range. Note that each value in the domain is also known as an input value, or independent variable, and is often labeled with the lowercase letter 𝑥. Each value in the range is also known as an output value, or dependent variable, and is often labeled lowercase letter 𝑦.

 

A function is a specific type of relation in which each input value has one and only one output value. A function can be thought of as sort of mathematical machine. You enter the input, the machine follows a specific rule, and produces the output. 

Function Notation

The notation 𝑦=𝑓⁡(𝑥) defines a function named 𝑓. This is read as ‶𝑦 is a function of 𝑥″  or "y equals f of x". The letter 𝑥 represents the input value, or independent variable. The letter 𝑦, or 𝑓⁡(𝑥), represents the output value, or dependent variable. Note that these are the most common notations used but other letters can also be used. 

To read function notation out loud, state the letter representing the function name followed by the word "of" and the input value, such as pronouncing ƒ(x) as "f of x."

ƒ(x) = 4x +1  read as "f of x equals 4x plus 1.

g(t) = 3t  read as "g of t equals 3t

II. Solving Function Problems

To solve a function problem, substitute the given input value for every instance of the variable in the equation and simplify the resulting expression to determine the output.

Example 1:

ƒ(x) = x + 5

Find the value of ƒ(10)

Here, the rule is add 5. Solve by simply replacing the x with the input value 10.

ƒ(10) = 10 + 5 = 15

III. Obtaining a Function from an Equation

1. The Conceptual Difference
Before you start the algebra, it is important to understand what you are trying to achieve:

• An Equation is a broad statement of equality between two expressions (e.g., x² + y² = 25). It describes a relationship, but one input (x) might lead to multiple outputs (y).
• A Function is a specific type of equation where every value of x (the input) is paired with exactly one value of y (the output).

2. Step-by-Step: Obtaining the Function

Step 1: Isolate the Dependent Variable
In most mathematics, y is the dependent variable. To find the function, you must solve the equation for y so that it stands alone on one side of the equals sign.

Example:
Start with the equation: 2x + y = 8

Subtract 2x from both sides:

y = -2x + 8

Step 2: Test for "Function-ness"
Once y is isolated, you must ensure that a single x value cannot produce two different y values. This is often checked using the Vertical Line Test if you are looking at a graph, or by checking for ± symbols in your algebra.

If you solve for y and end up with y = ±√x, the equation is not a function because one x (like 4) would result in two y values (2 and -2).

Step 3: Use Function Notation
If the equation passes the test, replace the variable y with the function notation ƒ(x). This signals that the value of the output depends directly on the input x.

Result:
ƒ(x) = -2x + 8

Graphing Functions

See: Lumen: Graphing Linear Functions

Reference

OpenStax: College Algebra

Lumen: Elementary Algebra: Intro to Functions

Khan: Functions and linear models

alt 0131 makes the function symbol

Measures of the Location of the Data

Quartiles, Percentiles, and Median

The Big Idea: Dividing Data

All three concepts are about locating positions within a dataset — they help you understand how values are distributed and where any particular value stands relative to the rest. The median is a midpoint of the distribution. 

I. Median

The median is the middle value of an ordered dataset. It splits data into two equal halves.

How to find it:

1. Sort your data from smallest to largest
2. If n is odd → the median is the middle value
3. If n is even → the median is the average of the two middle values

Example:

• Dataset: 3, 7, 8, 12, 15 → Median = 8 (middle value)
• Dataset: 3, 7, 8, 12 → Median = (7 + 8) / 2 = 7.5

 The median is also known as the 50th percentile, because 50% of values fall below it.

 

II. Quartiles

Quartiles divide ordered data into four equal parts (quarters).

Quartile	Symbol	Also Called	What it means
First Quartile	Q1	Lower Quartile	25% of data falls below this
Second Quartile	Q2	Median	50% of data falls below this
Third Quartile	Q3	Upper Quartile	75% of data falls below this

Example — Dataset: 2, 4, 6, 8, 10, 12, 14, 16

• Q2 (Median) = (8 + 10) / 2 = 9
• Q1 = median of the lower half {2, 4, 6, 8} = (4 + 6) / 2 = 5
• Q3 = median of the upper half {10, 12, 14, 16} = (12 + 14) / 2 = 13

*Note: We are using the most widely taught method of finding quartiles which is to exclude the median from the halves (exclusive method). There are other conventions where the median is included in both halves (inclusive methods).

How to find quartiles (Even number of data values)
1. Sort your data from smallest to largest
2. Find the median (second quartile) as the average of the two middle data values.

3. Because the median is excluded  when finding the quartiles, you will have an even number of data values in both halves of the data set. 

4. Find the median of the lower half (first quartile)

5. Find the median of the upper half (third quartile)  

Example data (10 values):
2, 3, 5, 6, 7, 9, 11, 13, 14, 18

1. The data is already sorted from smallest to largest
2. Median Q2: There are 10 values. The middle two are the 7 and 9. Q2 = (7 + 9)/2 =8.

3. Excluding the median, you have an even number of data values in both halves of the data set. 

2, 3, 5, 6, 7  |  9, 11, 13, 14, 18

4. The 5 is the middle value of the lower half so Q1 = 5

5. The 13 is the middle value of the upper half so Q3 = 13


Example data (9 values):

2, 3, 5, 6, 7, 9, 11, 13, 14

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III. Percentiles

Percentiles divide data into 100 equal parts. The pth percentile is the value below which p% of the data falls.

Formula (finding the position):

$$L = \frac{p}{100} \times n$$

Where p = percentile and n = number of data points. If L is a whole number, average the Lth and (L+1)th values. If L is a decimal, round up.

Example — Find the 30th percentile of: 5, 10, 15, 20, 25, 30, 35, 40 (n = 8)

$$L = \frac{30}{100} \times 8 = 2.4 → \text{round up to position 3}$$

The value at position 3 is 15, so the 30th percentile = 15.

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How They All Connect

0%          25%         50%         75%        100%
|___________|___________|___________|___________|
Min         Q1        Median        Q3         Max
           (25th      (50th        (75th
         percentile) percentile)  percentile)

• Median = Q2 = 50th percentile
• Q1 = 25th percentile
• Q3 = 75th percentile
• Every quartile is a percentile, but not every percentile is a quartile

use as an example

Claude AI

Lumen: Measures of the Location of the Data

Mathbitsnotebook: Quartiles

https://mathbitsnotebook.com/Algebra1/StatisticsData/STmmm.html

Frequency Polygons, and Time Series Graphs (Line Graphs)

Here we distinguish between two types of line graphs. While they both use points connected by line segments, they serve two completely different statistical purposes.

I. Frequency Polygons

A Frequency Polygon is used to show the distribution of a data set. It is essentially a "connected" version of a histogram. 

A. Constructing a Frequency Polygon

1. Sort Data (Same as histogram) Arrange your data in ascending order. This helps you quickly identify the minimum and maximum values to determine the spread of the data.

2. Define Bins (Same as histogram) Decide on the number of bins and calculate the bin width.

• Bin Width = (Maximum Value - Minimum Value) / Number of Bins.

• Note: Ensure your bin widths are consistent throughout the graph.

3. Create a Frequency Table with Midpoints List your intervals and tally the frequencies as you would for a histogram. However, for a frequency polygon, you must also calculate the midpoint for each interval.

• Formula: Midpoint = (Lower Boundary + Upper Boundary) / 2.

• Example: If a bin range is 100–110, the midpoint is 105.

4. Draw and Label the Axes 

• Horizontal Axis (X-axis): Label this with the measured variable (e.g., "Time in Hours"). Instead of marking just the boundaries, mark the midpoints you calculated in Step 3.

• Vertical Axis (Y-axis): Label this as "Frequency" or "Relative Frequency," ensuring the scale starts at zero.

5. Plot Points and Connect Instead of drawing vertical bars, you will create a line graph:

• Plot the Points: For each bin, place a dot at the intersection of its midpoint on the X-axis and its frequency on the Y-axis.

• Connect the Dots: Use a straight edge to connect the points in order from left to right.

Example:

S

One advantage of a frequency polygon is that it allows histogram-like data representation of two sets of data on the same graph. Two histograms on the same graph tend to shroud each other and make comparison more difficult, but two frequency polygons can be graphed together with much less interference

The figure below provides an example. The data come from a task in which the goal is to move a computer cursor to a target on the screen as fast as possible. On 20 of the trials, the target was a small rectangle; on the other 20, the target was a large rectangle. Time to reach the target was recorded on each trial. The two distributions (one for each target) are plotted together. The figure shows that, although there is some overlap in times, it generally took longer to move the cursor to the small target than to the large one.

II. Time Series Graphs

Though time series graphs look similar to frequency polygons, they display. Frequency polygons display the distribution of a data set (how often values fall into class intervals). Time series graphs shows how a specific variable changes over time. The horizontal axis is always time (days, months, years, etc.), and the vertical axis is the recorded value at each time point (such as temperature or sales).

A. Constructing a Time Series Graph

To construct a time series graph, we must look at both pieces of our paired data set using a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.


1. Organize Data Arrange your data points into a table in chronological order.

2. Draw and Label the Axes 

• Horizontal Axis (X-axis): This is the time variable.

• Vertical Axis (Y-axis): This is the quantity being measured. 

3. Plot the data points

• For each pair (time, value), find the position on the graph where that time on the x-axis meets that value on the y-axis.
• Place a small point or dot at each of these positions.

4. Connect the points

• Starting with the earliest time, connect consecutive points with straight line segments to show how the variable changes over time.

Example

The following data is about life expectancy in the U.S. from 1920-2000

Lumen: Histograms, Frequency Polygons and Time Series Graphs

ck-12: Frequency Polygons

https://www.socscistatistics.com/charts/frequencydistribution/calculator/

https://onlinestatbook.com/2/graphing_distributions/freq_poly.html

Histograms

I. Intro

A histogram is a specific type of data visualization that shows the distribution of a continuous variable. While it looks similar to a bar chart, it serves a very different purpose: instead of comparing discrete categories (like "Apples" vs. "Oranges"), it groups numerical data into ranges called bins. It is essentially a visual representation of frequency table for continuous data. 


The Horizontal Axis (x-axis): The Bins
This shows the variable you measured. It is what the data represents (for example, test scores, heights, ages) and is divided into contiguous groups called classes, intervals or bins that cover the range of the data. These bins are graphically shown as adjacent bars. Usually, every bar is equal (e.g., intervals of 10 units). The bars touch because there is no space between the end of one numerical range and the start of the next (note that the bars in a bar chart usually do not touch). 

The Vertical Axis (y-axis): The Frequency
The y-axis represents the count (or frequency). It tells you how many data points from your set fall into each specific bin. This can represent Frequency (total count) or Relative Frequency (the percentage of the total). The shape of the graph remains identical regardless of which you choose.

• Formula: Relative Frequency = Frequency/Total Number of Data Values 

II. Constructing a Histogram

1. Sort Data 

• Sort your data in ascending order. This identifies the range and makes counting easier. 

2. Define Bins

• Decide how many bins. Depending on the actual data distribution and the goals of the analysis, different bin widths may be appropriate. There are various useful guidelines and rules of thumb that we will go over below.
•  Once you know the number of bins, you need to determine the bin width. To ensure your bars are consistent, calculate the width of each interval: 
• Bid Width = Maximum Value - Minimum Value/Number of Bins. Adjust as needed. 

3. Create a Frequency Table

• List your intervals and tally how many data points fall into each. Add relative frequency column if your histogram is displaying relative frequency. 

4. Draw and Label the Axes

• Horizontal Axis (X-axis): Label this with what the data represents (e.g., "Height in Inches" or "Test Scores") and mark the bin boundaries.
• Vertical Axis (Y-axis): Label this as "Frequency" or "Relative Frequency." Ensure the scale starts at zero to avoid distorting the data.

5. Draw the Bars

• The Height: Matches the frequency of that bin. 
• The Width: Spans the entire interval on the X-axis

Example:

A manufacturer of AAA batteries had their quality control department test the lifespan of their batteries. Forty-two batteries were randomly selected and tested, with the number of hours they lasted listed below.

The Raw Data Set: 

108 125 137 110 167 158 142
168 163 121 134 146 135 163
148 153 169 154 156 142 160
147 119 124 145 167 161 155
138 126 149 168 151 129 157
115 124 165 152 159 144 163

Step 1: Sort Data

First, we take the scattered numbers and arrange them in ascending order. This allows us to easily find the Minimum (108) and Maximum (169) values to determine the spread. 

Sorted: 

108, 110, 115, 119, 121, 124, 124, 125, 126, 129, 134, 135, 137, 138, 142, 142, 144, 145, 146, 147, 148, 149, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 163, 163, 163, 165, 167, 167, 168, 168, 169

Step 2: Define Bins

We need to group these into intervals (bins). Let's aim for 7 bins.

Range = Largest value - Smallest value = 169 - 108 = 61

Bin Width: 61 / 7 = 8.7

To keep it simple for the graph, we’ll round the bin width to 10 and start our first bin at 100.

Step 3: Create a Frequency Table

Now we tally the sorted data into our defined intervals .

Step 4: Draw and Label the Axes

• Horizontal Axis (x axis): Label as "Time (Hours) and mark the bin boundaries (100 to 170).
• Vertical Axis (y axis): Label as "Number of Batteries" and add the range (0 to 12).

Step 5: Draw the Bars

Each bar is drawn to its specific frequency. Notice that the bars touch each other, indicating a continuous scale of data. The highest frequency occurs in the 160–169 bin with 11 values.

Gemini AI

Perplexity AI

Lumen: Histograms, Frequency Polygons, and Time Series Graphs

Wikipedia: Histogram

Understand Frequency Tables, Cumulative & Relative Frequency

CK-12: Understand and Interpret Frequency Tables and Histograms

Social Science Statistics Histogram & Frequency Table Maker

Stem-and-Leaf Graphs (Stemplots)

A stem-and-leaf plot (or stem-and-leaf display) is a unique way to organize numerical data so that you can see the overall shape of the distribution while still keeping every individual data point visible. It is essentially a hybrid between a table and a histogram.

To construct a stem-and-leaf display, the observations must first be sorted in ascending order
Here is the sorted set of data values that will be used in the following example:

44, 46, 47, 49, 63, 64, 66, 68, 68, 72, 72, 75, 76, 81, 84, 88, 106

Next, it must be determined what the stems will represent and what the leaves will represent. Typically, the leaf contains the last digit of the number and the stem contains all of the other digits. In the case of very large numbers, the data values may be rounded to a particular place value (such as the hundreds place) that will be used for the leaves. The remaining digits to the left of the rounded place value are used as the stem. In this example, the leaf represents the ones place and the stem will represent the rest of the number (tens place and higher).

The stem-and-leaf display is drawn with two columns separated by a vertical line. The stems are listed to the left of the vertical line. It is important that each stem is listed only once and that no numbers are skipped, even if it means that some stems have no leaves. The leaves are listed in increasing order in a row to the right of each stem.

The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down 50  instead of 500) while others may indicate that something unusual is happening.

Lumen: Stem-and-Leaf Graphs (Stemplots)

Wikipedia: Stem-and-leaf display

https://www.mathsisfun.com/data/stem-leaf-plots.html

Gemini AI

Intro to Descriptive Statistics

I. Overview

Descriptive statistics involves summarizing and organizing data from a sample to reveal its key characteristics, without making broader inferences about a population. It doesn’t try to make predictions or reach conclusions about a larger population (that’s inferential statistics); it simply describes exactly what you have in front of you.

Main Types

1. Measures of Central Tendency
This tells you where the "middle" of the data sits.

• Mean: The mathematical average.
• Median: The middle value when the data is lined up in order.
• Mode: The most frequently occurring value.

2. Measures of Variability (Spread)
This tells you how "stretched out" or "clustered" your data is.

• Range: The distance between the highest and lowest values.
• Standard Deviation: How much the data points typically deviate from the mean.
• Variance: The squared version of standard deviation, representing the degree of spread

3. Frequency Distribution
This is often shown as a table or a graph (like a histogram) that shows how often each individual value occurs. It helps you see the "shape" of your data—whether it's a perfect bell curve or skewed to one side.

II. Graphs

Descriptive statistics often uses graphs as a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data clusters and where there are only a few data values.

Some of the types of graphs that are used to summarize and organize data are:

• the dot plot
• the bar graph
• the histogram
• the stem-and-leaf plot
• the frequency polygon (a type of broken line graph)
• the pie chart
• the box plot.

Gemini AI

Lumen: Intro to Descriptive Statistics