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Chapter 3: Problem 41

Use the intercept method to graph each equation. $$ -9 x+4 y=9 $$

Short Answer

Expert verified
Plot points (-1, 0) and (0, 9/4) and connect with a straight line.

Step by step solution

01

Identify the X-Intercept

To find the x-intercept, set \( y = 0 \) in the equation. The equation becomes:\[-9x + 4(0) = 9 \]This simplifies to:\[-9x = 9 \]Solve for \( x \) by dividing both sides by \(-9\):\[x = -1 \]So, the x-intercept is \((-1, 0)\).
02

Identify the Y-Intercept

To find the y-intercept, set \( x = 0 \) in the equation. The equation becomes:\[-9(0) + 4y = 9 \]This simplifies to:\[4y = 9 \]Solve for \( y \) by dividing both sides by \(4\):\[y = \frac{9}{4} \]So, the y-intercept is \((0, \frac{9}{4})\).
03

Plot the Intercepts

Plot the points we found: the x-intercept \((-1, 0)\) and the y-intercept \((0, \frac{9}{4})\) on the coordinate plane.
04

Draw the Line

Draw a straight line through the points \((-1, 0)\) and \((0, \frac{9}{4})\). This represents the graph of the equation \(-9x + 4y = 9\).

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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 is a key point where a graph crosses the x-axis. To find this value for any linear equation, set the y-value to zero, since all points on the x-axis have y-coordinates of zero. This will simplify the equation, allowing you to solve for x. For the equation \[-9x + 4y = 9\]setting \( y = 0 \) gives:\[-9x + 4(0) = 9\]Simplifying, we find:\[-9x = 9\]By dividing both sides by \(-9\), we solve for x:\[x = -1\]Thus, the x-intercept for the given equation is the point \((-1, 0)\).
Remember, the x-intercept will always be of the form \((x, 0)\). Don't forget this simple swap of setting \(y = 0\) to quickly identify this intercept.
Y-Intercept
The y-intercept is where the graph crosses the y-axis. Every point on the y-axis has an x-coordinate of zero, which simplifies finding the y-intercept. When x is zero, solve for y:\[-9(0) + 4y = 9\]This reduces to:\[4y = 9\]By dividing both sides by 4, the solution for y is:\[y = \frac{9}{4}\]Therefore, the y-intercept for this equation is \( (0, \frac{9}{4}) \).The y-intercept will always be in the form \((0, y)\). Don't forget to set \( x = 0 \) when solving for the y-intercept. It's a straightforward way to find that crossing point on the coordinate plane.
Graphing Linear Equations
Graphing linear equations using the intercept method is efficient. Once you have the x-intercept and y-intercept, you simply plot these two points. For the equation \(-9x + 4y = 9\)we have found:
  • X-Intercept: \((-1, 0)\)
  • Y-Intercept: \((0, \frac{9}{4})\)
With these points plotted on the graph:1. Place a dot at each intercept.2. Use a ruler to draw a straight line through them.This line represents the graph of the linear equation, showing all possible solutions.
Graphing aids in visualizing how the line extends in both directions, creating a continuous set of solutions. Always ensure your line is straight and connects the intercepts accurately for a precise graph.

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Most popular questions from this chapter

Explain how to decide which side of the boundary line to shade when graphing a linear inequality in two variables.

Explain the difference between the graph of the solution set of \(x+1>8,\) an inequality in one variable, and the graph of \(x+y>8,\) an inequality in two variables.

Find an equation of the line that passes through \((2,5)\) and is parallel to the line \(y=4 x-7 .\) Write the equation in slope-intercept form.

Find the slope of each line. See Examples 4 and \(5 .\) $$ y=8 $$

For each pair of equations, determine whether their graphs are parallel, perpendicular, or neither. See Example 6 $$ \begin{aligned} &3 x=5 y-10\\\ &5 x=1-3 y \end{aligned} $$
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