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Two equations are equivalent if they have the same solutions. Solving a Linear Equation An equation is a statement in which two expressions are equal. A linear equation in one variable is an equation that can be written in the form ax = b where a and b are constants and a 0. A number is a solution of an equation if the statement is true when the number is substituted for the variable. For instance, the equations x – 4 = 1 and x = 5 are equivalent because both have the number 5 as their only solution. The transformations, or changes, on the following slide produce equivalent equations and can be used to solve an equation.
Solving a Linear Equation TRANSFORMATIONS THAT PRODUCE EQUIVALENT EQUATIONS ADDITION PROPERTY OF EQUALITY SUBTRACTION PROPERTY OF EQUALITY MULTIPLICATION PROPERTY OF EQUALITY DIVISION PROPERTY OF EQUALITY Add the same number to both sides: If a = b, then a + c = b + c. Subtract the same number from both sides: If a = b, then a – c = b – c. Multiply both sides by the same nonzero number: If a = b and c 0, then ac = bc. Divide both sides by the same nonzero number: If a = b and c 0, then a c = b c.
Your goal is to isolate the variable on one side of the equation. Write original equation. Subtract 9 from each side. Simplify. SOLUTION Solving an Equation with a Variable on One Side Solve + 9 = 15. 3737 x + 9 = 15 3737 x = 6 3737 x Multiply each side by, the reciprocal of. 7373 3737 = 6 x 7373 (6) = 6 x 14 The solution is 14. Substitute 14 for x. Solution checks. + 9 = 15 3737 ( x ) C HECK ? 15 = 15 14
Example Fractions (Multiply by LCD)
You do: (multiply by LCD)
Solve 5n + 11 = 7n – 9. 5n + 11 = 7n – 9 5n + 11 = 2n – 9 5n + 20 = 2n 10 = n Divide each side by 2. Add 9 to each side. Subtract 5 n from each side. Write original equation. SOLUTION Solving an Equation with a Variable on Both Sides The solution is 10. Check this in the original equation.
You do: 6x – 4 = 2x + 10
Solve 4(3x – 5) = –2(–x + 8) – 6x. 4(3x – 5) = –2(–x + 8) – 6x 12x – 20 = 2x –16 – 6x 12x – 20 = – 4x – 16 Divide each side by 16. Combine like terms. Distributive property Write original equation. SOLUTION Using the Distributive Property 16x – 20 = –16 Add 4 x to each side. 16x = 4 Add 20 to each side. x =x = 1414 The solution is. Check this in the original equation. 1414
You do: 2(x + 2) = 3(x – 8)
REAL ESTATE A real estate broker’s base salary is $18,000. She earns a 4% commission on total sales. How much must she sell to earn $55,000 total? SOLUTION Verbal Model Labels Total income Commission rate Total income = 55,000 Base salary = 18,000 Commission rate = 0. 04 (dollars) (percent in decimal form) Base salary + Using Linear Equations in Real Life = Total sales Total sales = x (dollars) Algebraic Model 18,000 + 0. 04x 55,000 =
REAL ESTATE A real estate broker’s base salary is $18,000. She earns a 4% commission on total sales. How much must she sell to earn $55,000 total? SOLUTION 18,000 + 0. 04x 55,000 = 37,000 = 0. 04x 925,000 = x Write linear equation. Subtract 18,000 from each side. Divide each side by 0.04. The broker must sell real estate worth a total of $925,000 to earn $55,000. Writing and Using a Geometric Formula
You do: A car salesperson’s base salary is $21,000. She earns a 5 % commission on sales. How much must she sell to earn $65,000 total?
Example The state sales tax in Pennsylvania is 0.06 (or 6%). If your total bill at the music store included $1.32 in tax, how much did the merchandise cost?
You have a 3 inch by 5 inch photo that you want to enlarge, mat, and frame. You want the width of the mat to be 2 inches on all sides. You want the perimeter of the framed photo to be 44 inches. By what percent should you enlarge the photo? SOLUTION Verbal Model Labels Perimeter + Perimeter = 44 (inches) 2 Width = 2 Length Width = 4 + 3x Length = 4 + 5x 44 = 2(4 + 3x) + 2(4 + 5x) Algebraic Model Writing and Using a Geometric Formula
44 = 2(4 + 3x) + 2(4 + 5x) Write linear equation. Distribute and combine like terms. 44 = 16 + 16x Subtract 16 from each side. 28 = 16x Divide each side by 16. 1. 75 = x You have a 3 inch by 5 inch photo that you want to enlarge, mat, and frame. You want the width of the mat to be 2 inches on all sides. You want the perimeter of the framed photo to be 44 inches. By what percent should you enlarge the photo? SOLUTION You should enlarge the photo to 175% of its original size. Writing and Using a Geometric Formula
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