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Chapter 3: Problem 56

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ -\frac{9}{4} y=x $$

Short Answer

Expert verified
The x- and y-intercepts are both at (0, 0). The line can be graphed through the origin.

Step by step solution

01

Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \( y \). Starting with the equation: \( -\frac{9}{4} y = x \) Set \( x = 0 \): \( -\frac{9}{4} y = 0 \) This simplifies to: \( y = 0 \) So, the y-intercept is at the point (0, 0).
02

Find the x-intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x \). Starting with the equation: \( -\frac{9}{4} y = x \) Set \( y = 0 \): \( -\frac{9}{4} (0) = x \) This simplifies to: \( x = 0 \) So, the x-intercept is also at the point (0, 0).
03

Graph the equation

To graph the equation \( -\frac{9}{4} y = x \), note that both the x- and y-intercepts are at the origin (0,0). This tells us the line passes through the origin. This is a linear equation, so the graph is a straight line. You can find another point to help draw the line. For example, when \( x = 4 \): \( -\frac{9}{4} y = 4 \) Solving for \( y \): \( y = -\frac{4}{9} \times 4 \) \( y = -\frac{16}{9} \) Plot the points (0,0) and (4, -\frac{16}{9}) and draw the line through them.

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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 found by determining where the line crosses the x-axis. This happens when the value of y is zero. To find the x-intercept for any equation:
  • Set y to 0.
  • Solve for x.
In the given equation \(-\frac{9}{4} y = x\), set y to 0: \(-\frac{9}{4} y = x \rightarrow -\frac{9}{4}(0) = x \rightarrow x = 0\)

The x-intercept here is at the point (0,0). This means the line crosses the x-axis at the origin.
y-intercept
The y-intercept is where the line crosses the y-axis. At this point, the value of x is zero. Here's how you can find the y-intercept:
  • Set x to 0.
  • Solve for y.
For the same equation \(-\frac{9}{4} y = x\), set x to 0: \(-\frac{9}{4} y = x \rightarrow -\frac{9}{4} y = 0 \rightarrow y = 0\)

The y-intercept is also at the point (0,0). This means the line also crosses the y-axis at the origin.
graphing equations
Graphing a linear equation involves plotting points and drawing the line that passes through them. For the equation \(-\frac{9}{4} y = x\), here are some steps:
  • Identify the x- and y-intercepts.
  • Plot these intercept points on a graph.
  • Find additional points if needed and plot them.
  • Draw a straight line through all the points.
Since both intercepts are at (0,0), it passes through the origin. Finding another point helps ensure accuracy. For example, when x = 4: \(-\frac{9}{4} y = 4 \rightarrow y = -\frac{16}{9}\). Plot (0,0) and (4, -16/9), then draw the line through these points.
solving linear equations
Solving linear equations means finding the values of x and y that make the equation true. These steps usually help:
  • Isolate the variable you are solving for.
  • Apply arithmetic operations to simplify.
  • Ensure both sides of the equation remain balanced.
In the given equation \(-\frac{9}{4} y = x\), when solving for y, set x to any number and solve. For instance, if x = 4, the process would be: \(-\frac{9}{4} y = 4 \rightarrow y = -\frac{4}{9} \times 4 \rightarrow y = -\frac{16}{9}\)

By following these steps, you can find any point on the line represented by the linear equation.

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Most popular questions from this chapter

Three points that lie on the same straight line are said to be collinear. Consider the points \(A(3,1), B(6,2),\) and \(C(9,3) .\) Use the slope formula to determine whether the points \((1,-2),(3,-1),\) and \((5,0)\) are collinear.

If a line has slope 0.2, then any line parallel to it has slope _____, and any line perpendicular to it has slope _____.

Use your knowledge of the slopes of parallel and perpendicular lines. Is the figure with vertices at \((-11,-5),(-2,-19),(12,-10),\) and \((3,4)\) a parallelogram? Is it a rectangle? (Hint: A rectangle is a parallelogram with a right angle.)

Find an equation of the line that satisfies the given conditions. (a) Write the equation in slope-intercept form. (b) Write the equation in standard form. Through \((-2,-2) ;\) parallel to \(-x+2 y=10\)

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ 4 x-3 y=8 $$
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