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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
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