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Master One-Variable Equations: Step-by-Step Solutions

Learn to solve equations with rational coefficients, two-step equations, equations with variables on each side, and multi-step equations following clear examples and explanations. Enhance your algebra skills now!

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Master One-Variable Equations: Step-by-Step Solutions

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  1. Topic 6 Equations with one variable and inequalities

  2. 8.2.1 Solve Equations with Rational Coefficients • To solve an equation when the coefficient is a rational number, multiply each side by the multiplicative inverse (the reciprocal) of the fraction.

  3. Example 1: Solve x = 16 .x = 16 .Multiply each side by the multiplicative inverse of x = 28

  4. 8.2.2 Solve Two-Step Equations • A two-step equation contains two operations. To solve a two-step equation, undo each operation in reverse order. • (Do PEDMAS backwards.)

  5. Example 1 Solve 2 + 6 = 14. – 6 – 6 Subtract 6 from both sides to cancel 2 = 8 Simplify. = Divide both sides by 2 to cancel. = 4 Example 2 Solve 5 = 8 – 2– 7. 5 = 6 – 7 Simplify. +7 + 7 Add 7 to both sides to cancel 12 = 6Simplify. = Divide both sides by 6 to cancel. 2 =

  6. 8.2.3 Write two-step equations More than (add) Sum of (add) Less than (subtract) Difference between (subtract) Product of (multiply) Times (multiply) Twice (multiply by 2) Quotient of (divide) Half (divide by 2)

  7. Example 2 • (4x –y)9 = • [4x + (-y)]9 = • (4x)9 + (-y)9 = • 36x + (-9y) = • 36x –9y Example1 5 (x + 3) = 5 · x + 5 · 3 = 5x + 15

  8. 8.2.4 Solve Equations with Variables on Each Side Some equations, like 3x – 9 = 6x, have variables on each side of the equals sign. Use addition or subtraction to write an equivalent equation with the variables on one side of the equals sign. Then solve the equation.

  9. Example1 Solve 3x - 9 = 6x. All variables must be on same side of equation 3x - 3x - 9 = 6x - 3x Subtract 3x from both sides -9 = 3x Simplify by combining like terms. -3 = x Divide each side by 3. Example 2 Solve 4a - 7 = 5 - 2aAll variables must be on same side of equation 4a + 2a - 7 = 5 - 2a + 2aAdd 2a to both sides 6a - 7 = 5 Simplify by combining terms with like variables 6a - 7 + 7 = 5 + 7 Add 7 to both sides 6a = 12 Simplify a = 2Divide both sides by 6

  10. 8.2.5 Solve Multi-Step Equations • Equations should be as simplified as possible before you solve for the variable.

  11. Example1 2(4a - 5) = 30 8a - 10 = 30 Distributive Property 8a - 10 + 10 = 30 + 10 Add 10 to both sides 8a = 40 Simplify a = 5 Divide both sides by 8. Example 2 Alan has 3 paperback books and 4 hardcover books. Each hardcover book is worth $11 more than each paperback book. If the value of all of his books is $79, what is the cost of one paperback book? (Use the variable p for the price of a paperback book.) 3p + 4(p + 11) = 79 Write the equation. 3p + 4p + 44 = 79 Distributive Property 7p + 44 = 79 Simplify. 7p + 44 + (-44) = 79 + (-44) Subtract 44 from each side 7p = 35 Simplify. p = 5 Divide each side by 7

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