4.5 Graphing Linear Equations

• A linear equation can be written in the form Ax + By = C.

• This is called the standard form of a linear equation.

• A ≥ 0, A and B are not both zero, and A, B, and C are integers whose greatest common factor is 1.

Identify Linear Equations• Determine whether each equation is a linear

equation. If so, write the equation in standard form.

a. y = 5 – 2x

y + 2x = 5 – 2x + 2x

2x + y = 5

The equation is now in standard form whereA = 2, B = 1, and C = 5. This is a linear equation.

Identify Linear Equations• Determine whether each equation is a linear

equation. If so, write the equation in standard form.

b. 2xy – 5y = 6

Since the term 2xy has two variables, the equation cannot be written in the form Ax + By = C. Therefore, this is not a linear equation.

Identify Linear Equations• Determine whether each equation is a linear

equation. If so, write the equation in standard form.

c. 3x + 9y = 15Since the GCF of 3, 9, and 15 is not 1, the

equation is not written in standard form. Divide each side by the GCF.

3x + 9y = 153(x + 3y) = 15x + 3y = 5

The equation is now in standard form where A = 1, B = 3, and C = 5.

Identify Linear Equations• Determine whether each equation is a linear

equation. If so, write the equation in standard form.

d. 1/3 y = -1To write the equation with integer coefficients,

multiply each term by 3.1/3 y = -13(1 /3 ) y = 3(-1)y = -3

The equation y = -3 can be written as 0x + y = -3. Therefore, it is a linear equation in standard form where A = 0, B = 1, and C = -3.

Graph Linear Equations

• The graph of a linear equation is a line.

• The line represents all the solutions of the linear equation.

• Also, every ordered pair on this line satisfies the equation.

Graph by Making a Table• Graph x + 2y = 6.

• In order to find values for y more easily, solve the equation for y.

x + 2y = 6

x + 2y – x = 6 – x

2y = 6 – x

y = 3 – ½ x

Graph by Making a Table

Intercepts

• Since two points determine a line, a simple method of graphing a linear equation is to find the points where the graph crosses the x-axis and the y-axis.

• The x-coordinate of the point at which it crosses the x-axis is the x-intercept, and the y-coordinate of the point at which the graph crosses the y-axis is called the y-intercept.

Graph Using Intercepts

• Determine the x-intercept and y-intercept of 3x + 2y = 9. Then graph the equation.

• To find the x-intercept, let y = 0.

3x + 2y = 9

3x + 2(0) = 9

3x = 9

x = 3

Graph Using Intercepts

• Determine the x-intercept and y-intercept of 3x + 2y = 9. Then graph the equation.

• To find the y-intercept, let x = 0.

3x + 2y = 9

3(0) + 2y = 9

2y = 9

y = 4.5

Graph Using Intercepts• The x-intercept is 3, so the graph intersects the x-axis

at (3 , 0). The y-intercept is 4.5, so the graph intersects the y-axis is (0 , 4.5). Plot these points. Then draw a line that connects them.