 # Goal: I will solve linear equations in one variable. ❖ Linear equations in one variable with one solution, infinitely many solutions, or no solutions.

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Goal: I will solve linear equations in one variable. ❖ Linear equations in one variable with one solution, infinitely many solutions, or no solutions. ❖ Linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms.

1) GOAL :Get the variable on one side of the equation. 2) You always perform the same operation to both sides of an equation. 3) To undo an operation, you perform its opposite operation to both sides of the equation. 4) You should undo addition or subtraction before you undo multiplication or division. Solving 2 Step Equations - Let’s Review

2x + 4 = 10 Get the variable alone. - 4 - 4 Subtract 4 from both sides. 2x = 6 +4 and -4 cancel out __ __ 2 2 Divide both sides by 2 x = 3 Show your check. 2(3) +4 = 10 Finish your check. 6 + 4 = 10 10 = 10 2 divided by 2 cancel to 1 Solving 2 Step Equations

– 4 = 8 Get the variable alone. 3 ( ) = 3 12 Multiply both sides by 3. x = 36 3 divided by 3 cancel to 1 x3x3 x3x3 +4 +4 Add 4 to both sides. x3x3 = 12 + 4 and -4 cancel out. – 4 = 8 36 3 12 – 4 = 8 8 = 8 Show your check. Finish your check. Solving 2 Step Equations

Try These Examples 1. 2x – 5 = 17 2. 3x + 7 = 25 3. x - 3 = 8 5 4. x + 4 = 9 9

Don’t Call Me Stupid… D – Distribute C – Combine like terms M – Move the variable to one side S – Solve

You need to get the variables on one side of the equation. It does not matter which variable you move. Try to move the one that will keep your variable positive.

1) Solve 8x - 9 = -3x + 2 + 3x + 3x 11x – 9 = 2 + 9 + 9 11x = 11 11 11 x = 1 8(1) - 9 = -3(1) + 2 1. Draw “the river” 2. Add 3x to both sides Simplify Add 9 to both sides Simplify Divide both sides by 11 Simplify Check your answer

2) Solve 3x + 2 = 4x - 1 - 3x 2 = x - 1 + 1 + 1 3 = x 3(3) + 2 = 4(3) - 1 9 + 2 = 12 - 1 1. Draw “the river” 2. Subtract 3x from both sides 3. Simplify 4. Add 1 to both sides 5. Simplify 6. Check your answer Multi-Step Equations: Get Variables on One Side

Try these on your own… 2x – 14 = 3x – 13 Multi-Step Equations: Get Variables on One Side

Try these on your own… 5y + 7 = 13 + 3y Multi-Step Equations: Get Variables on One Side

Try these on your own… 11x + 30 = 5x – 42 Multi-Step Equations: Get Variables on One Side

3) Solve 4 = 7x - 3x 4 = 4x 4 4 1 = x 4 = 7(1) - 3(1) 1.Draw “the river” – Notice the variables are on the same side! 2.Combine like terms 3.Divide both sides by 4 4.Simplify 5.Check your answer Solve Equations by Combining Like Terms

18 = 8x + x 2x – 4x + 2 = 22 Solve Equations by Combining Like Terms

-x + 7x + 1 = 37 10x – 6x = 2x + 14 Solve Equations by Combining Like Terms

3) Solve -7(x - 3) = -7 -7x + 21 = -7 - 21 - 21 -7x = -28 -7 -7 x = 4 -7(4 - 3) = -7 -7(1) = -7 1.Draw “the river” 2.Distribute 3.Subtract 21 from both sides 4.Simplify 5.Divide both sides by -7 6.Simplify 7.Check your answer Solve Equations by Using the Distributive Property

What is the value of x if 3(x + 4) = 2(x - 1)? Answer Now 1. -14 2. -13 3. 13 4. 14 Solve Equations by Using the Distributive Property

Give it a shot… 3(x + 3) = 18 -4(y – 2) = 24 Solve Equations by Using the Distributive Property

4(2x – 5) + 4 = x + 5 Solve Equations by Using the Distributive Property

Putting it all together 3(x – 4) + 4x = 12 + x Remember: D C M S !!!

Practice… Don’t Call Me Stupid Write the letters you use to solve the problem. Which letter will always be used? 1) 3x – 14 = 46 1/3x – 4.1 = 6 3(3x – 3) = 27 Remember: D C M S !!!

4) 12.2x – 16.7 = 1.4x + 37.3 5) 9(x – 3) + 4 = 49 + 3x Remember: D C M S !!!

Challenge! What is the value of x if -8(x + 1) + 3(x - 2) = -3x + 2? Remember: D C M S !!!

Balancing Scales 1.These scales are all currently balanced. You must choose a number to fill into the boxes in each problem that will keep them balanced. 2.Whatever number you choose for a problem, you may only use that number.

Special Case #1 6) 2x + 5 = 2x - 3 -2x 5 = -3 This is never true! No solutions 1.Draw “the river” 2.Subtract 2x from both sides 3.Simplify Remember: D C M S !!!

SPECIAL CASES 2x + 3 = 2x + 7

SPECIAL CASES 3(x + 4) = 3x + 11

Special Case #2 7) 3(x + 1) - 5 = 3x - 2 3x + 3 – 5 = 3x - 2 3x - 2 = 3x – 2 -3x -3x -2 = -2 This is always true! Infinite solutions or identity 1.Draw “the river” 2.Distribute 3.Combine like terms 4.Subtract 3x from both sides 5.Simplify Remember: D C M S !!!

SPECIAL CASES 2x + 3 = 2x + 3

SPECIAL CASES -2(x + 3) = -2x - 6

What is the value of x if -3 + 12x = 12x - 3? 1. 0 2. 4 3. No solutions 4. Infinite solutions Answer Now Remember: D C M S !!!

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