Core Focus on Linear Equations Lesson 1.7 Core Focus on Linear Equations Solutions to Linear Equations

Warm-Up Solve each equation for x. 1. 33 = 5(x + 8) + 3 x = 2 2. 2m + 3(m – 8) = 1 3. 3x + 10 = 9x – 26 x = 2 m = 5 x = 6

Solutions to Linear Equations Lesson 1.7 Solutions to Linear Equations Determine if a linear equation in one variable has no solution, one solution or infinitely many solutions.

Explore! What Works? Linear equations can have one solution, no solution, or infinitely many solutions (meaning any number would make the equation true). Step 1 Choose an equation out of the green box above. Solve the equation for x. What do you notice about the solution? Do you think your equation has one solution, no solution or infinitely many solutions? Why? Step 2 Find a partner in the room who solved the same equation as you did in Step 1. Do you agree on how many solutions your equation has? If not, determine whose reasoning is better. Step 3 With your partner, solve another equation from the box. How many solutions do you think this equation has based on your work? Why? Step 4 Solve the last equation in the box. There is one of each type (one, no or infinitely many solutions). Based on your work, decide which equation is each type. Step 5 How can you tell how many solutions a linear equation has based on your work solving the equation? Write two or three summary statements about your findings.

Example 1 Solve for x. Describe the number of solutions. Group like terms. Move variables to the same side of the equation. The ending statement is true for all values of x, therefore the equation has infinitely many solutions. 6x + 12 = 2x – 5 + 4x + 17 6x + 12 = 6x + 12 –6x –6x a 12 = 12 a

Extra Example 1 Solve 3x + 1 = 4x − 3 for x. Describe the number of solutions. x = 4; there is one solution

Example 2 Solve for x. Describe the number of solutions. Group like terms (3x + 7x). Distribute and simplify. Move variables to the same side of the equation. The ending statement is false for all values of x, therefore the equation has no solution. 3x + 4 + 7x = 5(2x + 1) 10x + 4 = 10x + 5 –10x –10x a 4 = 5 a Four does not equal 5 so there are no values of x that make this equation true.

Extra Example 2 Solve −2x + 12 + 10x = 4(2x + 3) for x. Describe the number of solutions. 12 = 12; there are infinitely many solutions

Example 3 Solve for x. Describe the number of solutions. Distribute. Group like terms and simplify. Move variables to the same side of the equation. Subtract 11 from both sides of the equation. Divide both sides by 2. The ending statement has one solution. The only value that makes the equation true is x = 3. 8(x + 2) – 5 = 6(x – 1) + 23 8x + 16 – 5 = 6x – 6 + 23 8x + 11 = 6x + 17 –6x –6x 2x + 11 = 17 –11 –11 2x = 6a 2 2a x = 3 a

Extra Example 3 Solve 3(x − 6) + 12 = 6(x + 1) − 3x for x. Describe the number of solutions. −6 = 6; there are no solutions

Determining the Number of Solutions to a Linear Equation A linear equation in one variable has… One solution if, when solved, the variable is equal to one number. No solution if, when solved, the equation makes a false statement (e.g., 4 = 5). Infinitely many solutions if, when solved, the equation makes a statement that is true no matter what value of the variable is tested (e.g., 12x = 12x).

Communication Prompt What does it mean when an equation has infinitely many solutions?

Exit Problems Solve each equation. There may be one solution, no solutions, or infinitely many solutions. 2x + 6 = 6x  22 2x + 8 = 2x  3 3(x – 4) = x + 2x – 12 x = 7; one solution 8 = 3; no solutions −12 = −12; infinitely many solutions