Chapter 1 Section 1 Lesson: Solving Linear Equations

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1 Introduction Linear equations are the simplest types of equations to solve. In a linear equation, all variables are to the first power only. All linear equations in one variable can be reduced to the form mx + b 0, where x is the variable and m and b are constants. Despite their simplicity, linear equations are very useful in modeling many events that occur in the world. In a later section, we ll see how solving a linear equation is equivalent to finding where a line crosses the x -axis in a graph (this is why such equations are called linear). Solving Linear Equations There are three basic steps to solve a linear equation: 1. Get all of the terms with the variable you re solving for on one side of the equal sign and all other terms on the other side of the equal sign.. Factor out the variable you re solving for, if necessary.. Divide both sides of the equal sign by the variable s coefficient. Example A Solve: 5x 60. The x term is already all by itself on the left side of the equation. Just divide both sides by 5 and we re done: 5x x 1 Example B Solve: x + 0. The term doesn t have our variable, so we move it to the other side of the equal sign. This is easily accomplished by subtracting from both sides of the equation: x + 0 x Next divide both sides by, the coefficient of the variable: x x x Page 1 of 1

2 Extended Example 1a Solve: 9x Hint: Subtract 5 from both sides of the equation. Step1: 9x x Hint: Divide both sides by 9. 9 x 9 9 x 9 Extended Example 1b Solve: 5x 1 0. Hint: Add 1 to both sides of the equation. Step1: 5x x Hint: Divide both sides by 5. 5 x 5 5 x 5 Extended Example 1c Solve: 6x 1 1. Hint: Add 1 to both sides of the equation. 6x x 6 Hint: Divide both sides by 6. Step : 6 x Hint: Simplify the fraction. 6 1 x 6 1 x Page of 1

3 Example C Solve: 6y + 9 8y 5. We ll move the two terms involving y to the right side of the equation and the two terms that don t involve y to the left side of the equation. This is accomplished by subtracting 6y from both sides, and adding 5 to both sides: 6y + 9 8y 5 6y + 5 6y y Finally, divide both sides by, the coefficient of the variable: 14 y 7 y Note that the answer can also be written as y 7. Example D Solve: 1x 4 7x + x 5 + 4x 8. First combine all like terms, on both sides of the equal sign: 1x 4 7x + x 5+ 4x 8 5x 6x 1 Now we ll move the two terms involving x to the right side of the equation, and the two terms that don t involve x to the left side of the equation. This is accomplished by subtracting 5x from both sides, while adding 1 to both sides: 5x 6x 1 5x + 1 5x x Page of 1

4 Extended Example a Solve: 4x + 7 4x 6 5x + + 7x 9. Hint: Combine all like terms, on both sides of the equal sign. 4x + 7 4x 6 5x + + 7x 9 0x + 1 1x 6 Hint: Subtract 1x from both sides and subtract 1 from both sides. Step : 0x + 1 1x 6 1x 1 1x 1 8x 7 Hint: Divide both sides by 8. 8 x x 8 Extended Example b Solve: 1x + 6x 8 + x x + 8x 5. Hint: Combine all like terms, on both sides of the equal sign. 1x + 6x 8+ x x + 8x 5 10x 6 7x 8 Hint: Subtract 7x from both sides and add 6 to both sides. Step : 10x 6 7x 8 7x + 6 7x + 6 x Hint: Divide both sides by. x x Page 4 of 1

5 Extended Example c Solve: 0x + 5 6x 8 + 8x x x +. Hint: Combine all like terms, on both sides of the equal sign. 0x + 5 6x 8+ 8x x 11+ 8x + x 7x + 1 Hint: Subtract 7x from both sides and add to both sides. Step : x 7x + 1 7x + 7x + 15x 15 Hint: Divide both sides by x x 1 Example E Solve: A(4 A) 9 A( A ). First we need to eliminate the parentheses (using the distributive property) so that we can then combine the like terms on both sides of the equation. Notice on the right side how the A is distributed through the parentheses, sign and all: 4A A 9 A + 6A. This isn t a linear equation, since we have A squared. However, in this case we can simply add A to each side of the equation to eliminate the nonlinear terms: 4A A 9 A + 6A + A + A 4A 9 + 6A Subtract 6A to get all the A terms on the left side of the equation: 4A 9 + 6A 6A 6A A 9 Finally, divide both sides by, the coefficient of the variable: A 9 9 A Page 5 of 1

6 Extended Example a Solve: y(5 + 6 y) 8 y(4 y). Hint: Use the distributive property to eliminate the parentheses. 5y + 6y 8 8y + 6y Hint: Subtract 6y from both sides. Step : 5y + 6y 8 8y + 6y 6y 6y 5y 8 8y Hint: Add 8y to both sides. Step : 5y 8 8y + 8y + 8y 1y 8 Hint: Divide both sides by 1. 1y y y 1 Extended Example b Solve: 6 y(y 10) + 1 y(6+ 9 y). Hint: Use the distributive property to eliminate the parentheses. 18y 60y + 1 1y + 18y Hint: Subtract 18y from both sides. Step : 18y 60y + 1 1y + 18y 18y 18y 1 60y 1y Hint: Add 60y to both sides. continued Page 6 of 1

7 Extended Example b, continued Step : 1 60y 1y + 60y + 60y 1 7y Hint: Divide both sides by y y y 7 Extended Example c Solve: 4 y( y 10) y( + y) + 6. Hint: Use the distributive property to eliminate the parentheses. 4y 40y 6y + 4y + 6 Hint: Subtract 4y from both sides. Step : 4y 40y 6y + 4y + 6 4y 4y 40y 6y + 6 Hint: Add 6y to both sides. Step : 40y 6y y + 6 y 4y 6 Hint: Divide both sides by 4. Step 4: 4y y Hint: Simplify the fraction. 6 1 y y 17 Page 7 of 1

8 Example F Solve: x 4 x The easiest way to deal with such equations is to first cancel all the denominators. To accomplish this, all we need to do is to multiply both sides of the equation by the least common multiple (LCM) of all the denominators. For this equation, the LCM is 5, or 5 7: x 4 x x x x 4 15x Get the x terms together by subtracting 14x from both sides: 14x 4 15x 14 x 14 x 4 x Extended Example 4a Solve: 7 x 5 5 x. 6 Hint: Multiply both sides by the LCM of the denominators, 6, and distribute it to cancel the denominators. 7x 5 5x x x 6 6 7x 5 5x 6 7x 15 10x Hint: Subtract 7x from both sides. Step : 7x 15 10x 7 x 7x 15 x Hint: Divide both sides by. continued Page 8 of 1

9 Extended Example 4a, continued 15 x x 5 5 x Extended Example 4b Solve: 9 x 7 x Hint: Multiply both sides by the LCM of the denominators, 10, and distribute it to cancel the denominators. 9x 7x x x x x x + 5x Hint: Subtract 18x from both sides. Step : 18x + 5x 18x 18x 17x Hint: Divide both sides by x x x Page 9 of 1

10 Extended Example 4c Solve: 11 x 8 x Hint: Multiply both sides by the LCM of the denominators, 1, and distribute it to cancel the denominators. 11x 8 x x x x x x + 9x Hint: Subtract 9x and from both sides. Step : x + 9x 9x 9x 1x Hint: Divide both sides by 1. 1x x 1 1 x 1 Example G Solve: z z 5. This isn t a linear equation, but once we clear the denominators it will be. The LCM of all the denominators is 15z, or 5 z, so we multiply both sides of the equation by 5 z : z 5 z z z 5 5 z 6 5 z 5 5 z 4 z z z 65 1z continued Page 10 of 1

11 Example G, continued 65 1z 65 1z z z 1 Extended Example 5a Solve: 7 5. x x Hint: Multiply both sides by the LCM of the denominators, x, and distribute it to cancel the denominators. 7 5 x x x x x 7 x x 5 x x x 7 x x 5 x x 1 5x 19 5x Hint: Divide both sides by x 19 5x x x 5 Page 11 of 1

12 Extended Example 5b 5 5 Solve:. x x 4 Hint: Multiply both sides by the LCM of the denominators, 4x, and distribute it to cancel the denominators x 4x x x 4 4x 5 4x 4x 5 x x 4 x 5 4 x 4 x 5 x x x 5x Hint: Divide both sides by 5. 5x 5x x 5 5 x 5 Extended Example 5c Solve: x 10 x Hint: Multiply both sides by the LCM of the denominators, 10x, and distribute it to cancel the denominators x + 10x 5x 10 x 10x 1 10x 10x 5 + 5x 10 x 5x 1 10 x x x 10 x 6 + x 5 Hint: Subtract 6 from both sides. continued Page 1 of 1

13 Extended Example 5c, continued Step : 6+ x x 1 Hint: Divide both sides by. x 1 x 1 1 x Example H Solve: 0.57x x First, clear the decimals by multiplying both sides of the equation by 100 : ( x ) ( x ) x 89 x + 11 Now we ll move the two terms involving x to the left side of the equation, and the two terms that don t involve x to the right side of the equation. This is accomplished by subtracting x from both sides and adding 89 to both sides: 57x 89 x + 11 x + 89 x x x x 17 End of Lesson Page 1 of 1

Solving Algebraic Equations

Lesson 4. Solving Algebraic Equations 3 3 3 3 3 8 8 4 Add 3 to both sides. Divide both sides by. 4 gives the solution of the equation 3. Check: Substitute 4 for x into the original equation. 3 4 3 When