Exploring Linear Equations: Chapter 3 in Algebra

1. Where we’ve been… • Today, we take our first step into Chapter 3: Linear Functions – which is fundamentally at the of Algebra • Before we take our first step, let’s go back and see what objectives we accomplished in Chapter 2. • In Chapter 2, we… • Solved equations by using the 4 basic operations • Solved one-step, two-step, and multi-step equations • Used formulas to solve real world problems

2. Now, in Chapter 3, our objective will be to… • 1. Identify linear equations • 2. Identify x- and y- intercepts • 3. Graph linear equations using the x- and y-intercepts or a table of values with 4 domain values

3. 3.1 Graphing Linear Equations Our objective: To identify a linear equation in Standard Form To identify the X-intercept and Y-intercept of a linear equation

4. Identifying a Linear Equation Ax + By = C or Ax – By = C • The exponent of each variable is 1. • The variables are added or subtracted. • A or B can equal zero. • A> 0 (cannot have a negative coefficient) • Besides x and y, other commonly used variables are m and n, a and b, and r and s. • There are no radicals (sq. root symbols) in the equation. • Every linear equation graphs as a line.

5. Examples of linear equations Equation is in Ax + By =C form Rewrite with both variables on left side … x + 6y =3 B=0 … x + 0 y =1 Multiply both sides of the equation by -1 … 2a – b = -5 Multiply both sides of the equation by 3 … 4x –y =-21 2x + 4y =8 6y = 3 – x x = 1 -2a + b = 5

6. Examples of Nonlinear Equations The following equations are NOT in the standard form of Ax + By =C: 4x2 + y = 5 xy + x = 5 s/r + r = 3 The exponent is 2 There is a radical in the equation Variables are multiplied Variables are divided (can’t have a variable in the denominator)

7. x and y -intercepts • The x-intercept is the point where a line crosses the x-axis. The general form of the x-intercept is (x, 0). The y-coordinate will always be zero. • The y-intercept is the point where a line crosses the y-axis. The general form of the y-intercept is (0, y). The x-coordinate will always be zero.

8. Finding the x-intercept • For the equation 2x + y = 6, we know that y must equal 0. What must x equal? • Plug in 0 for y and simplify. 2x + 0 = 6 2x = 6 x = 3 • So (3, 0) is the x-intercept of the line.

9. Finding the y-intercept • For the equation 2x + y = 6, we know that x must equal 0. What must y equal? • Plug in 0 for x and simplify. 2(0) + y = 6 0 + y = 6 y = 6 • So (0, 6) is the y-intercept of the line.

10. To summarize…. • To find the x-intercept, plug in 0 for y. • To find the y-intercept, plug in 0 for x.

11. x-intercept: Plug in y = 0 x = 4y - 5 x = 4(0) - 5 x = 0 - 5 x = -5 (-5, 0) is the x-intercept y-intercept: Plug in x = 0 x = 4y - 5 0 = 4y - 5 5 = 4y = y (0, ) is the y-intercept Find the x and y- interceptsof x = 4y – 5

12. x-intercept Plug in y = 0 g(x) = -3x - 1 0 = -3x - 1 1 = -3x = x ( , 0) is the x-intercept y-intercept Plug in x = 0 g(x) = -3(0) - 1 g(x)= 0 - 1 g(x) = -1 (0, -1) is the y-intercept Find the x and y-interceptsof g(x) = -3x – 1* *g(x) is the same as y

13. x-intercept Plug in y = 0 6x - 3y = -18 6x -3(0) = -18 6x - 0 = -18 6x = -18 x = -3 (-3, 0) is the x-intercept y-intercept Plug in x = 0 6x -3y = -18 6(0) -3y = -18 0 - 3y = -18 -3y = -18 y = 6 (0, 6) is the y-intercept Find the x and y-intercepts of 6x - 3y =-18

14. x y Find the x and y-intercepts of x = 3 • x-intercept • Plug in y = 0. There is no y. Why? • x = 3 is a verticalline so x always equals 3. • (3, 0) is the x-intercept. • y-intercept • A vertical line never crosses the y-axis. • There is no y-intercept.

15. x y Find the x and y-intercepts of y = -2 • y-intercept • y = -2 is a horizontal line so y always equals -2. • (0,-2) is the y-intercept. • x-intercept • Plug in y = 0. y cannot = 0 because y = -2. • y = -2 is a horizontal line so it never crosses the x-axis. • There is no x-intercept.