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Solving Equations: Examples and Guided Practice

This article provides examples and guided practice on solving equations, including factoring, multiplying with the least common denominator, and solving multi-step problems. It also emphasizes the importance of checking solutions.

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Solving Equations: Examples and Guided Practice

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  1. x 2 1 = + x – 2 x –2 5 x 1 2 5(x – 2) + 5(x – 2) = 5(x – 2) x – 2 5 x –2 25(x – 2) 5(x – 2) x 5(x – 2) + = x – 2 x – 2 5 5x + x – 2 = 10 EXAMPLE 2 Multiply by the LCD x 2 1 Solve . Check your solution. = + x – 2 x –2 5 SOLUTION Write original equation. Multiply by LCD, 5(x – 2). Multiply and divide out common factors. Simplify.

  2. 6x – 2 = 10 6x = 12 x = 2 x The solution appears to be 2, but the expressions and are undefined when x= 2. So, 2 is an extraneous solution. x – 2 2 x –2 There is no solution. ANSWER EXAMPLE 2 Multiply by the LCD Combine like terms. Add 2 to each side. Divide each side by 6.

  3. 3 8 + 1 = x – 7 (x – 2)(x – 7) 8 3 (x – 2)(x – 7) (x – 2)(x – 7) = (x – 2)(x – 7) + 1 x – 7 (x – 2)(x – 7) 8 (x – 2)(x – 7) 3(x – 2)(x – 7) = + (x – 2)(x – 7) (x – 2)(x – 7) x – 7 EXAMPLE 3 Factor to find the LCD 3 8 Solve . Check your solution. + 1 = x – 7 x2 – 9x + 14 SOLUTION Write each denominator in factored form. The LCD is (x – 2)(x – 7).

  4. x2 – 6x + 8 = 8 x2 – 6x = 0 x(x – 6) = 0 The solutions are 0 and 6. ANSWER EXAMPLE 3 Factor to find the LCD 3(x – 2) + (x2 – 9x + 14) = 8 x = 0 or x – 6 = 0 x = 0 or x = 6

  5. ? ? 3 8 3 8 + 1 = + 1 = 0 – 7 02 – 9 0+ 14 6 – 7 62 – 9 6+ 14 4 4 = –2 –2 = 7 7 EXAMPLE 3 Factor to find the LCD CHECK Ifx = 0: Ifx = 6:

  6. EXAMPLE 4 Solve a multi-step problem PAINT MIXING You have an 8 pint mixture of paint that is made up of equal amounts of yellow paint and blue paint. To create a certain shade of green, you need a paint mixture that is 80% yellow. How many pints of yellow paint do you need to add to the mixture? SOLUTION Because the amount of yellow paint equals the amount of blue paint, the mixture has 4 pints of yellow paint. Let prepresent the number of pints of yellow paint that you need to add.

  7. 4 + p = 0.8 8 + p EXAMPLE 4 Solve a multi-step problem STEP 1 Write a verbal model. Then write an equation. STEP 2 Solve the equation. Write equation.

  8. You need to add 12 pints of yellow paint. ANSWER 4 + p = 0.8 8 + p 4 + 12 ? = 0.8 8 + 12 EXAMPLE 4 Solve a multi-step problem 4 + p = 0.8(8 + p) Cross products property 4 + p = 6.4 + 0.8p Distributive property 0.2p = 2.4 Rewrite equation. p = 12 Solve for p. CHECK Write original equation. Substitute 12 forp.

  9. ? 16 = 0.8 20 0.8 = 0.8 EXAMPLE 4 Solve a multi-step problem Simplify numerator and denominator. Write fraction as decimal. Solution checks.

  10. n 22 –10 = – 1 = n – 11 n2 – 5n – 66 a ANSWER ANSWER a = –4: for Examples 2, 3, and 4 GUIDED PRACTICE Solve the equation. Check your solution. – 12 a 1 = + 3 a + 4 a + 4

  11. You need to add 8 pints of yellow paint. ANSWER for Examples 2, 3, and 4 GUIDED PRACTICE WHAT IF?In Example 4, suppose you need a paint mixture that is 75% yellow. How many pints of yellow paint do you need to add to the mixture?

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