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Chapter 3: Problem 35

Find the intercepts. Then graph. $$ 3 x+2 y=12 $$

Short Answer

Expert verified
The x-intercept is (4, 0) and the y-intercept is (0, 6).

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. \[ 3x + 2(0) = 12 \] This simplifies to: \[ 3x = 12 \] Divide both sides by 3: \[ x = 4 \] So the x-intercept is (4, 0).
02

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. \[ 3(0) + 2y = 12 \] This simplifies to: \[ 2y = 12 \] Divide both sides by 2: \[ y = 6 \] So the y-intercept is (0, 6).
03

- Plot the intercepts

Plot the points (4, 0) and (0, 6) on the coordinate plane.
04

- Draw the line

Draw a straight line through the points (4, 0) and (0, 6) to graph the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept
The x-intercept of a linear equation is the point where the line crosses the x-axis. To find this point, set the variable y to 0 and solve the equation for x. In the given problem, you start with the equation: \( 3x + 2y = 12 \). By replacing y with 0, the equation becomes: \[ 3x + 2(0) = 12 \]. Simplifying this, we get \[ 3x = 12 \]. When we divide both sides by 3, we find \[ x = 4 \]. Thus, the x-intercept is (4, 0). This point is where the line touches the x-axis.
y-intercept
The y-intercept of a linear equation is where the line crosses the y-axis. To find it, set x to 0 and solve the equation for y. Using the same problem: \( 3x + 2y = 12 \), replace x with 0, the equation becomes: \[ 3(0) + 2y = 12 \]. Simplifying this, we get \[ 2y = 12 \]. When we divide both sides by 2, we find \[ y = 6 \]. This makes the y-intercept (0, 6). This point is where the line meets the y-axis.
graphing linear equations
Once you have the intercepts, you can graph a linear equation on the coordinate plane. Start by plotting the x-intercept and y-intercept. For this exercise, plot (4, 0) and (0, 6). These intercepts are key points that help in drawing the line. After plotting these points, draw a straight line that passes through both. This line represents all the solutions to the equation \( 3x + 2y = 12 \). Each point on this line satisfies the given linear equation.
coordinate plane
The coordinate plane is a two-dimensional surface where you can graph equations. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes divide the plane into four quadrants. Each point on the plane can be described using an ordered pair (x, y). The first number represents the position along the x-axis, and the second number represents the position along the y-axis. By plotting points like the x-intercept (4, 0) and y-intercept (0, 6) on the coordinate plane, you can easily visualize and graph linear equations.

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