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

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