I. Two-Step Equations
A two-step equation is an algebraic equation with two operations in it that requires two steps to solve. As discussed in the intro to equations section, the goal is to isolate the variable on one side of the equation.
To isolate the variable, we have to undo the involved operations by using their inverse operations (opposite operations) and solve for the variable. In general, we want to perform addition or subtraction first, then perform multiplication or division.
To isolate the variable, we have to undo the involved operations by using their inverse operations (opposite operations) and solve for the variable. In general, we want to perform addition or subtraction first, then perform multiplication or division.
So the basic steps to solve two-steps equations are to balance both sides of the equation using following rules:
- Undo the addition by subtracting both sides with the same number.
- Undo subtraction by adding both sides with the same number.
- Undo the multiplication by dividing both sides with the same number.
- Undo the division by multiplying the same number to both sides.
Example: Solve 2x + 1 = 5
Step 1: Subtract 1 from both sides.
2x + 1 - 1 = 5 - 1
2x = 4
Step 2: Divide both sides by 2
2x/2 = 4/2
x = 2
Example: Solve 3x/4 - 2/3 = 7/3
Step 1: Add 2/3 to both sides
3x/4 - 2/3 + 2/3 = 7/3 + 2/3
3x/4 = 9/3
Step 2: Multiply 3/4 to both sides
(4/3) 3x/4 = 9/3 (4/3)
x = 4
Example:
II. Solving Multi-Step Linear Equations
The multi-step equations are algebraic equations that require multiple steps to solve. Solving multi-step equations in algebra is similar to solving one-step and two-step equations, but the process is a little lengthy as there are multiple steps involved.A. Equations With Variables On Both Sides
Some equations may have the variable on both sides of the equal sign, as in this equation:
4x − 6 = 2x + 10
To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It doesn’t matter which term gets moved, 4x or 2x, however, to avoid negative coefficients, you can move the smaller term.
B. Equations With Fractions
Sometimes, you will encounter a multi-step equation with fractions. If you prefer not working with fractions, you can use the multiplication property of equality to multiply both sides of the equation by a common denominator of all of the fractions in the equation (Least Common Denominator). This will clear all the fractions out of the equation.
1. Find the Least Common Denominator (LCD): Identify all the denominators in the equation. The LCD is the smallest multiple that all the denominators divide into evenly.
2. Multiply Every Term by the LCD: Multiply both sides of the equation (and every individual term within them) by the LCD. This will cancel out all the denominators.
3. Solve the Resulting Equation: After clearing the fractions, you’ll have an equation with whole numbers. Solve this equation using the standard steps (combine like terms, isolate the variable).
4. Check Your Solution: Substitute your answer back into the original equation (with fractions) to verify that it is correct.
D. Classify Solutions to Linear Equations
To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It doesn’t matter which term gets moved, 4x or 2x, however, to avoid negative coefficients, you can move the smaller term.
Sometimes, you will encounter a multi-step equation with fractions. If you prefer not working with fractions, you can use the multiplication property of equality to multiply both sides of the equation by a common denominator of all of the fractions in the equation (Least Common Denominator). This will clear all the fractions out of the equation.
1. Find the Least Common Denominator (LCD): Identify all the denominators in the equation. The LCD is the smallest multiple that all the denominators divide into evenly.
2. Multiply Every Term by the LCD: Multiply both sides of the equation (and every individual term within them) by the LCD. This will cancel out all the denominators.
3. Solve the Resulting Equation: After clearing the fractions, you’ll have an equation with whole numbers. Solve this equation using the standard steps (combine like terms, isolate the variable).
4. Check Your Solution: Substitute your answer back into the original equation (with fractions) to verify that it is correct.
Example with variable in denominator
C. Equations With Decimals
Sometimes, you will encounter a multi-step equation with decimals. If you prefer not working with decimals, you can use the multiplication property of equality to multiply both sides of the equation by a a factor of 10 that will help clear the decimals. See the example below.
D. Classify Solutions to Linear Equations
There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that don’t have any solutions, and even some that have an infinite number of solutions. The case where an equation has no solution is illustrated in the next examples.
No Solution
This is not a solution! You did not find a value for x. Solving for x the way you know how, you arrive at the false statement 4=54=5. Surely 4 cannot be equal to 5!
This is not a solution! You did not find a value for x. Solving for x the way you know how, you arrive at the false statement 4=54=5. Surely 4 cannot be equal to 5!
Infinite Number of Solutions
Example
Solve: 5x + 3 -4x = 3 + x
Combine like terms
x + 3 = 3 + x
Subtract 3 from both sides
x + 3 - 3 = 3 - 3 + x
x = x
E. Equation Word Problems
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