# 1.9 – Solving Multistep Linear Equations

Objectives

• Isolate the variables in an equation or inequality by identifying the operations involved.
• Collect like terms to simplify the equation or inequality.
• Remember the order of operations for solving an equation or inequality in one variable in the proper sequence.
• Identify equations that have no solution or an infinite number of solutions.
• Ask the questions necessary to turn real-life problems into mathematical sentences and equations in one variable.

Key Terms

• Distributive Property – The rule that if a, b, and c are numbers or expressions, then a(b + c) = ab + ac.
• Example
• 3(x + 4) – 2x = 5
• Step 2: Distribute (using multiplication)
• 3x + 12 – 2x = 5
• Step 3: Collect like terms
• (3x – 2x) + 12 = 5
• Step 4: Simplify
• x + 12 = 5
• Step 5: Solve
• X = -7

• Infinite – Without end or limit; going on forever; impossible to count.
• Like Terms – Terms in an algebraic expression that have the same variables raised to the same powers.
• 5ab and -22ab are like terms because they have the same variables in each term.
• 8y and -5x are not like terms because they have different variables.

• Linear Equations – Equations whose graphs are straight lines. A linear equation can be written in the form y = mx + b.

Notes

• To solve these types of equations, you have to “undo” the variable expression. First you have to isolate the variable term, and then you have to isolate the variable itself.
• To undo (cancel) addition (positives), you must subtract terms on BOTH sides.
• To undo (cancel) subtraction (negatives), you must add the terms to BOTH sides.
• To undo (cancel) multiplication, you must divide terms or coefficients on BOTH sides.
• To undo (cancel) division, you must multiply terms or coefficients on BOTH sides.

• No Solution
• When an equation has no solution, there is no value for x that would make the equation true.
• Example: for x = x + 7, no matter what value you use for x, you will always get the untrue statement 0 = 7.
• Subtract x on both sides: 0 = 7
• 7 does not equal zero!  So, this is untrue; therefore, no solution!

• Infinite Solutions
• If you try to solve an equation that has an infinite number of solutions, you will end up with an equation that is always true, no matter what the value of x is.
• Example: if x = x is the answer (or if two sides of the equation are identical), every solution is possible; so, infinite solutions is the answer.
• Start with the equation: 3 (x + 2) = 3x + 6
• Distribute 3 into the parenthesis: 3x + 6 = 3x + 6
• Notice how the two sides are identical!
• Subtract 6 from both sides: 3x = 3x
• Divide by 3 on both sides: x = x
• So, for ANY value you substitute into x, both sides will ALWAYS be equal.

• Example: 3x + 4 = 13
• Step 1: Subtract 4 from both sides to isolate the variable term
• 3x = 9
• Step 2: Divide both sides by 3 to isolate the variable
• x = 3

• Example: 3x – 4 = 17
• Step 1: Add 4 to both sides to isolate the variable term
• 3x = 21
• Step 2: Divide both sides by 3 to isolate the variable
• x = 7

• Example
• $\frac{1}{2}(4x + 4) = x +\frac{1}{2}(4 - 6x) + 8$
• Step 2: Distribute (multiply 1/2 into each term inside the parenthesis)
• $2x + 2 = x + 2 - 3x + 8$
• Step 3: Collect like terms
• $2x + 2 = (x - 3x) + (2 + 8)$
• Step 4: Simplify (combine like terms)
• $2x + 2 = -2x + 10$
• Step 5: Add 2x to both sides
• $4x + 2 = 10$
• Step 6: Subtract 2 from both sides
• $4x = 8$
• Step 7: Divide both sides by 4 and write the answer
• $x = 2$