Linear Equations Sections 10.1 – 10.3

I can recognize linear equations. I can solve equations of the form x + B = C. I can solve equations of the form Ax = C I can solve equations of the form Ax + B = C

Write an Equation for the Following I added $30 to my bank account and my balance is $330. What did I start with? I paid $50 for 10 bags of cherries. How much did each bag cost? Parking costs a flat rate of $3.00 plus $2.00 per hour. I spent $13.00. How long was I parked?

These Are Linear Equations x + 30 = 330 10x = 50 2x + 3 = 13 Any equation that can be written in the form Ax + B = C, where A, B, C are real numbers.

NOT Linear equations x2 + 5x -3 = 0 |x - 3| = 7 1/x = 12 √x = 25.

Linear or Nonlinear? 5 = 2x 3 – s = ¼ 3 – t2 = ¼ 50 = ¼ r2

Goal: Solve Linear Equations We have simplified expressions with one or sometimes more than one variable. Today we are going to learn how to solve linear equations in one variable.

This means: what value of x makes the sentence true? Try This Problem I deposited $30 into my bank account and my new balance was $330. What did I start with? How did you figure this out? Try to solve this equation: x + 3 = 7. This means: what value of x makes the sentence true?

Simplifying Expressions When you simplify expressions, you only have one side of a scale. The weight cannot change. 2(x+3) 2x+6

Solving Equations When you solve equations, you have both sides of the scale! You can change the weight, but the scale must balance! x + 3 = 7 x + 3 - 3 = 7 - 3 x = 4

Inverse Operations To solve x + 3 = 7, you subtract 3 from both sides. To solve x - 3 = 7, what do you do? Addition and subtraction are inverse operations. We always use inverse operations to solve equations.

Always check your answer! More Examples If -x = a, then x = -a 5 – k = 12 5m + 4 = 6m y + 2/3 = ½ ½ x – 5 = -1/2 x + 2 6m – 5m = m (like terms) Always check your answer!

Don’t forget to check your answer! Try Some x – 17 = 25 12 – r = 7 t – ½ = 3/4 Don’t forget to check your answer!

Use the distributive property! Simplify First Solve 5t – 4t + 6 = 9 Simplify first, then solve Solve 4x + 6 + 2x – 3 = 9 + 5x – 4 Solve: 3(2+5x) – (1+14x) = 6 5t – 4t = t t + 6 = 9 Use the distributive property!

What About Multiplication? If I paid $50 for 10 bags of cherries, how much did each bag of cherries cost? How did you figure this out? Solve: 5x = 60

Try Some More Solve -25p = 50 Solve 2m = 15 Solve -6x = 14.

Fractions I In algebra, instead of writing x ¥ 3, we write x/3. This is consistent with the notion that division is multiplication by the reciprocal. To solve x/3 = 10, how do we undo the division?

Fractions II What about the equation 2/3 x = 6? To divide by 2/3, multiply by the reciprocal. Since (3/2) £ (2/3) = 1, we now have: x = (3/2) £ 6. Work this out, what do we get? x = 9

Let’s do Another Together Solve: 7/5 x = 9/4 What is the reciprocal of 7/5? Multiply both sides by 5/7. What do we have on the left? For the right, what is 5/7 times 9/4? What is our solution? 5/7 Just x The product of a number with its reciprocal is 1! 45/28 x = 45/28

Try These Solve y/12 = 5 Solve 1/3 z = 19 Solve 6/7 t = 9/5

Simplify First Just like before, sometimes you have to simplify before solving! Solve: 5x + 6x = 9 Solve 7x – 2x = -25

Let’s Put These Together! Parking costs a flat rate of $3.00 plus $2.00 per hour. I spent $13.00. How long was I parked? Answer the question with your partner. How did you solve this problem?

The Easiest Way To Do This The equation for this problem is: 2h + 3 = 13. The variable h stands for hours. First: subtract the 3 from both sides: 2h = 10 Second: divide both sides by 2: h = 5.

The order is the opposite! Another example The order is the opposite! Solve 3x -5 = 7 First: addition and subtraction Second: multiplication and division Add 5 to both sides 3x = 12 Divide both sides by 3 x = 4

Try These: Solve: 5x – 6 = 17 Solve: -4x + 2 = 9

You can solve linear equations using inverse operations. If you have to deal with both addition and multiplication, deal with addition first.