Linear Equations

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.

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.

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!

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

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.

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.

Linear Equations