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Chapter 7: Problem 18

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 4 y=3 x $$

Short Answer

Expert verified
Both intercepts are at (0,0); plot the origin and another point like (4,3), then draw the line.

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. Substituting x = 0 into the equation gives: 4 y = 3(0), which simplifies to 4 y = 0. Therefore, the y-intercept is y = 0.
02

- Find the x-intercept

To find the x-intercept, set the value of y to 0 in the equation. Substituting y = 0 into the equation gives: 4(0) = 3 x, which simplifies to 0 = 3 x. Therefore, the x-intercept is x = 0.
03

- Graph the equation

Plot the intercepts on the coordinate plane. Both the x-intercept and the y-intercept are at the origin (0,0). Since both intercepts are the same point, this indicates that the graph of the equation passes through the origin. For additional accuracy, choose another value for x or y and solve for the corresponding coordinate to plot another point. For example, let x = 4: 4 y = 3(4), therefore y = 3. Plotting the points (0,0) and (4,3) and drawing a line through them gives the graph of 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 line is the point where the line crosses the x-axis. To find this point in any linear equation, set y to 0 and solve for x. For instance, in the example equation 4y = 3x, setting y = 0 results in the equation 4(0) = 3x, simplifying to 0 = 3x. Solving for x, you find that x = 0.This means that the x-intercept is at the origin, (0,0). This is the point where the line touches the x-axis.
y-intercept
The y-intercept is found where the line crosses the y-axis. To determine this, you should set x to 0 and solve for y. In our given example, 4y = 3x, setting x = 0 leads us to 4(y) = 3(0), which simplifies to 4y = 0.Solving for y, you'll find that y = 0.Therefore, the y-intercept is also at the origin, (0,0). This point shows where the line interacts with the y-axis.
graphing linear equations
Graphing linear equations involves plotting points on a coordinate plane and drawing a line through them. The intercepts, both x and y, give key points on the graph. For our example, both intercepts are at the origin, (0, 0). After finding the intercepts, it's often helpful to find another point to ensure accuracy. For instance, if x = 4: 4y = 3(4), which simplifies to 4y = 12 and thus y = 3.Now, you can plot (4, 3) along with (0, 0) and draw a line through these points. This line represents the graph of the equation.
coordinate plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where you can plot points, lines, and curves. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0,0).Points on this plane are specified by ordered pairs (x, y), where x is the horizontal distance from the origin and y is the vertical distance. When graphing equations, you plot these ordered pairs on the plane to visualize the relationship the equation describes.

Understanding the coordinate plane is crucial for graphing linear equations effectively.

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