VIEWS: 20 PAGES: 26 POSTED ON: 6/29/2012
Linear Equations Sections 10.1 – 10.3 ►Ican recognize linear equations. ►Ican solve equations of the form x + B = C. ►Ican solve equations of the form Ax = C ►Ican solve equations of the form Ax + B = C Write an Equation for the Following ►Iadded $30 to my bank account and my balance is $330. What did I start with? ►Ipaid $50 for 10 bags of cherries. How much did each bag cost? ► Parkingcosts 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 ► Anyequation 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 ► Wehave 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 ►5 – k = 12 If -x = a, then x = -a ► 5m + 4 = 6m 6m – 5m = m (like terms) ►y + 2/3 = ½ Always check your answer! ►½ x – 5 = -1/2 x + 2 Try Some ►x – 17 = 25 ► 12 –r=7 ►t – ½ = 3/4 Don’t forget to check your answer! Simplify First ► Solve 5t – 4t + 6 = 9 ► Simplify first, then solve 5t – 4t = t t+6=9 ► Solve 4x + 6 + 2x – 3 = 9 + 5x – 4 ► Solve: 3(2+5x) – (1+14x) = 6 Use the distributive property! What About Multiplication? I paid $50 for 10 bags of cherries, how ► If 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 ► Inalgebra, 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? 5/7 ► Multiply both sides by 5/7. ► What do we have on the left? Just x The product of a number with its reciprocal is 1! ► Forthe right, what is 5/7 times 9/4? 45/28 ► What is our solution? x = 45/28 Try These ► Solve y/12 = 5 ► Solve 1/3 z = 19 ► Solve 6/7 t = 9/5 Simplify First like before, sometimes you have to ► Just simplify before solving! ► Solve: 5x + 6x = 9 ► Solve 7x – 2x = -25 Let’s Put These Together! ► Parkingcosts 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. subtract the 3 from both sides: ► First: 2h = 10 ► Second: divide both sides by 2: h = 5. Another example ► Solve 3x -5 = 7 The order is the opposite! ► First: addition and subtraction Add 5 to both sides 3x = 12 ► Second: multiplication and division 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.