Chapter 1 Section 1 Solving Linear Equations in One Variable

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1 Chapter Section Solving Linear Equations in One Variable

2 A linear equation in one variable is an equation which can be written in the form: ax + b = c for a, b, and c real numbers with a 0. Linear equations in one variable: x + 3 = (x ) = 8 3 Not linear equations in one variable: x + 3y = Two variables can be rewritten x + ( ) = 8. x + 5 = x 7 can be rewritten x + 5 = 7. 3 (x ) = 8 x is squared. 3x + 5 = x 7 Variable in the denominator Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company

3 A solution of a linear equation in one variable is a real number which, when substituted for the variable in the equation, makes the equation true. Example: Is 3 a solution of x + 3 =? x + 3 = Original equation (3) + 3 = Substitute 3 for x = False equation 3 is not a solution of x + 3 =. Example: Is 4 a solution of x + 3 =? x + 3 = (4) + 3 = = 4 is a solution of x + 3 =. Original equation Substitute 4 for x. True equation Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 3

4 Addition Property of Equations If a = b, then a + c = b + c and a c = b c. That is, the same number can be added to or subtracted from each side of an equation without changing the solution of the equation. Use these properties to solve linear equations. Example: Solve x 5 =. x 5 = x = + 5 x = 7 Original equation The solution is preserved when 5 is added to both sides of the equation. 7 is the solution. 7 5 = Check the answer. Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 4

5 Multiplication Property of Equations a b If a = b and c 0, then ac = bc and =. c c That is, an equation can be multiplied or divided by the same nonzero real number without changing the solution of the equation. Example: Solve x + 7 = 9. x + 7 = 9 x = 9 7 x = (x) = () x = 6 (6) + 7 = + 7 = 9 Original equation The solution is preserved when 7 is subtracted from both sides. Simplify both sides. The solution is preserved when each side is multiplied by. 6 is the solution. Check the answer. Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 5

6 To solve a linear equation in one variable:. Simplify both sides of the equation.. Use the addition and subtraction properties to get all variable terms on the left-hand side and all constant terms on the right-hand side. 3. Simplify both sides of the equation. 4. Divide both sides of the equation by the coefficient of the variable. Example: Solve x + = 3(x 5). x + = 3(x 5) Original equation x + = 3x 5 x = 3x 6 x = 6 x = 8 The solution is 8. Check the solution: (8) + = 3((8) 5) 9 = 3(3) Simplify right-hand side. Subtract from both sides. Subtract 3x from both sides. Divide both sides by. True Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 6

7 Example: Solve 3(x + 5) + 4 = (x + 6). 3(x + 5) + 4 = (x + 6) 3x = x 3x + 9 = x 3x = x 30 5x = 30 x = 6 The solution is 6. 3( 6 + 5) + 4 = ( 6 + 6) Original equation Simplify. Simplify. Subtract 9. Add x. Divide by 5. Check. 3( ) + 4 = (0) = True Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 7

8 Equations with fractions can be simplified by multiplying both sides by a common denominator. Example: Solve x + = ( x + 4) x + = 6 ( x + 4) Multiply by x + 4 = x + 8 Simplify. 3x = x + 4 Subtract 4. x = 4 Subtract x. ( 4) + = (( 4) + 4) = (8) = 3 3 The lowest common denominator of all fractions in the equation is 6. Check. True Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 8

9 Alice has a coin purse containing $5.40 in dimes and quarters. There are 4 coins all together. How many dimes are in the coin purse? Let the number of dimes in the coin purse = d. Then the number of quarters = 4 d. 0d + 5(4 d) = 540 0d d = 540 Linear equation Simplify left-hand side. 0d 5d = 60 Subtract d = 60 d = 4 There are 4 dimes in Alice s coin purse. Simplify right-hand side. Divide by 5. Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 9

10 The sum of three consecutive integers is 54. What are the three integers? Three consecutive integers can be represented as n, n +, n +. n + (n + ) + (n + ) = 54 3n + 3 = 54 Linear equation Simplify left-hand side. 3n = 5 Subtract 3. n = 7 Divide by 3. The three consecutive integers are 7, 8, and = 54. Check. Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 0

11 Digital Lesson Linear Equations in Two Variables

12 Equations of the form ax + by = c are called linear equations in two variables. y This is the graph of the equation x + 3y =. - (0,4) (6,0) x The point (0,4) is the y-intercept. The point (6,0) is the x-intercept. Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company

13 The slope of a line is a number, m, which measures its steepness. y m is undefined m = m = - m = 0 x m = - 4 Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 3

14 The slope of the line passing through the two points (x, y ) and (x, y ) is given by the formula y y m =, (x x ). x x The slope is the change in y divided by the change in x as we move along the line from (x, y ) to (x, y ). y (x, y ) (x, y ) x x change in x y y change in y x Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 4

15 Example: Find the slope of the line passing through the points (, 3) and (4, 5). Use the slope formula with x =, y = 3, x = 4, and y = 5. m = y y x x y 5 3 = = 4 = (4, 5) (, 3) x Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 5

16 A linear equation written in the form y = mx + b is in slope-intercept form. The slope is m and the y-intercept is (0, b). To graph an equation in slope-intercept form:. Write the equation in the form y = mx + b. Identify m and b.. Plot the y-intercept (0, b). 3. Starting at the y-intercept, find another point on the line using the slope. 4. Draw the line through (0, b) and the point located using the slope. Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 6

17 Example: Graph the line y = x 4.. The equation y = x 4 is in the slope-intercept form. So, m = and b = -4. y. Plot the y-intercept, (0, -4). 3. The slope is. m = change in y change in x 4. Start at the point (0, 4). Count unit to the right and units up to locate a second point on the line. The point (, -) is also on the line. = (0, -4) (, -) x 5. Draw the line through (0, 4) and (, -). Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 7

18 A linear equation written in the form y y = m(x x ) is in point-slope form. The graph of this equation is a line with slope m passing through the point (x, y ). Example: y The graph of the equation y 3 = - (x 4) is a line of slope m = - passing through the point (4, 3). 8 4 m = - (4, 3) 4 8 x Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 8

19 Example: Write the slope-intercept form for the equation of the line through the point (-, 5) with a slope of 3. Use the point-slope form, y y = m(x x ), with m = 3 and (x, y ) = (-, 5). y y = m(x x ) Point-slope form y y = 3(x x ) Let m = 3. y 5 = 3(x (-)) Let (x, y ) = (-, 5). y 5= 3(x + ) y = 3x + Simplify. Slope-intercept form Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 9

20 Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (-, 5). m = = - 6 = - 3 y y = m(x x ) Calculate the slope. Point-slope form y 3 = - (x 4) Use m = - and the point (4, 3). 3 3 y = - x + 3 Slope-intercept form 3 3 Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company 0

21 Two lines are parallel if they have the same slope. If the lines have slopes m and m, then the lines are parallel whenever m = m. y Example: The lines y = x 3 and y = x + 4 have slopes m = and m =. y = x + 4 (0, 4) x The lines are parallel. (0, -3) y = x 3 Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company

22 Two lines are perpendicular if their slopes are negative reciprocals of each other. If two lines have slopes m and m, then the lines are perpendicular whenever m = - or m m m = -. y = 3x Example: The lines y = 3x and y = - x + 4 have slopes 3 m = and m = -. 3 The lines are perpendicular. y (0, 4) (0, -) y = - x x Mathematical Applications by Harshbarger (8th ed) Copyright by Houghton Mifflin Company

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