Chapter 3 Solving Linear Equations 3 1 Solving

Chapter 3 Solving Linear Equations

3. 1 Solving Equations Using Addition and Subtraction n You can solve an equation by using the transformations below to isolate the variable on one side of the equation. When you rewrite an equation using these transformations, you produce an equation with the same solutions as the original equation. These equations are called equivalent equations.

n n n To change, or transform, an equation into an equivalent equation, think of an equation as having two sides that are “in balance. ” Any transformation you apply to an equation must keep the equation in balance. For example, if you subtract 3 from one side of the equation, you must also subtract 3 from the other side of the equation.

Transformations that produce Equivalent Equations Original Equation Add the same x – 3 = 5 # to each side Subtract the x + 6 = 10 same # to each side Simplify one x = 8 – 3 or both sides Interchange the sides 7=x Equivalent Equation

Example 1: Solve. n a) x – 5 = -13 n b) x – 9 = -17 n Example 2: Solve. n a) -8 = n – (-4) n b) -11 = n – (-2) n

n Linear Equations: The variable is raised to the first power and does not occur in a denominator, inside a square root, or inside absolute value symbols. Linear Equation Not a Linear Equation x+5=9 x 2 + 5 = 9 - 4 + n = 2 n – 6 |x + 3| = 7

n Example 3: Several record temperature changes have taken place in Spearfish, South Dakota. On January 22, 1943, the temperature in Spearfish fell from 54°F at 9: 00 am to -4°F at 9: 27 am. By how many degrees did the temperature fall?

n n n Example 4: Match the real-life problem with an equation. x – 4 = 16 x + 16 = 4 16 – x = 4 a) You owe $16 to your cousin. You paid x dollars back and now you owe $4. How much did you pay back? b) The temperature was x°F. It rose 16°F and is now 4°F. What was the original temperature? c) A telephone pole extends 4 feet below ground and 16 feet above ground. What is the total length x of the pole?

3. 2 Solving Equations using Multiplication and Division Original Equation Multiply each equation by the same nonzero # Divide each equation by the same nonzero # 4 x = 12 Equivalent Equation

n n n Example 1: Solve. a) - 4 x = 1 b) 7 n = - 35 n Example 2: Solve. a) n b) n

n Example 3: Solve. a) n b) n

n Properties of Equality: Addition Property: n Subtraction Property: n Multiplication Property: n Division Property: n

3. 3 Solving Multi-Step Equations n n n Solving a linear equation may require two or more transformations. Simplify one or both sides of the equation (if needed). Use the inverse operations to isolate the variable.

n Example 1: Solve. a) n b) n n Example 2: Solve. a) 7 x – 3 x – 8 = 24 n b) 2 x – 9 x + 17 = - 4 n

n Example 3: Solve. a) 5 x + 3(x + 4) = 28 n b) 4 x – 3(x – 2) = 21 n n Example 4: Solve. a) 4 x + 12(x – 3) = 28 n b) 2 x – 5(x – 9) = 27 n

n Example 5: Solve. a) n b) n n Example 6: A body temperature of 95°F or lower may indicate the medical condition called hypothermia. What temperature in the Celsius scale may indicate hypothermia?

n Example 7: The temperature within Earth’s crust increases about 30° Celsius for each kilometer of depth beneath the surface. If the temperature at Earth’s surface is 24°C, at what depth would you expect the temperature to be 114°C?

3. 6 n n Objective: To find exact and apporoximate solutions of equations that contain decimals. Round-off Error: Example 1: Solve the equation. Round to the nearest hundredth. a) 7. 23 x + 16. 51 = 47. 89 – 2. 55 x

n Example 1: Solve the equation. Round to the nearest hundredth. a) 7. 23 x + 16. 51 = 47. 89 – 2. 55 x n b) 6. 6(1. 2 – 7. 3 x) = 16. 4 x + 5. 8 n

n Example 2: Multiply the equation by a power of 10 to write an equivalent equation with integer coefficients. Solve the equation and round to the nearest hundredth. a) 3. 11 x – 17. 64 = 2. 02 x -5. 89 n b) 5. 8 + 3. 2 x = 3. 4 x – 16. 7 n

3. 7 Formulas and Functions n n Objective: To solve a formula for one of its variables and rewrite an equation in function form Formula: an algebraic expression that relates two or more real-life quantities.

Example 1: Use the formula for area of a rectangle – A = lw a) Find a formula for l in terms of A and w n b) Use the new formula to find the length of a rectangle that has an area of 35 sq. ft. and a width of 7 feet.

Example 2: Solve the temperature formula C = 5/9(F – 32) for F. Example 3: a) Solve the simple interest formula for r. b) Find the interest rate for an investment of $1500 that earned $54 in simple interest in one year

Example 3: Rewrite the equation 3 x + y = 4 so that y is a function of x. Example 4: a) Rewrite the equation 3 x + y = 4 so that x is a function of y. b) Use the result to find x when y = -2, -1, 0 and 1