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Chapter 3: Problem 42

Use the intercept method to graph each equation. $$ -5 x+4 y=15 $$

Short Answer

Expert verified
The graph intersects the x-axis at \((-3, 0)\) and the y-axis at \((0, 3.75)\).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation. This simplifies the equation to \(-5x + 4(0) = 15\), which further simplifies to \(-5x = 15\). Solving for \(x\), divide both sides by -5 to get \(x = -3\). So the x-intercept is \((-3, 0)\).
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation. This simplifies the equation to \(-5(0) + 4y = 15\), simplifying further to \(4y = 15\). Divide both sides by 4 to solve for \(y\), giving \(y = \frac{15}{4}\) or \(3.75\). So the y-intercept is \((0, 3.75)\).
03

Plot the intercepts on a graph

On a coordinate plane, plot the x-intercept at \((-3, 0)\) and the y-intercept at \((0, 3.75)\).
04

Draw the line through the intercepts

Using a ruler, draw a straight line passing through the points \((-3, 0)\) and \((0, 3.75)\). This line represents the graph of the equation \(-5x + 4y = 15\).

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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 graph is the point where the line crosses the x-axis. This happens when the value of y is zero. To find the x-intercept of any linear equation, set y to zero in the equation and solve for x.

For example, in the equation \(-5x + 4y = 15\), finding the x-intercept involves substituting y with zero, leading to \(-5x = 15\). Solving this gives \(x = -3\).

Therefore, the x-intercept is the point \((-3, 0)\). You can always check your work by plugging this intercept back into the original equation to see if it balances. Remember, anytime the x-coordinate is non-zero and the y-coordinate is zero, you've pinpointed an x-intercept on the graph.
y-intercept
The y-intercept is where a line crosses the y-axis. This is found by setting x to zero and solving for y in your equation.

In the equation \(-5x + 4y = 15\), make x zero, which simplifies the equation to \(4y = 15\). By solving for y, you get \(y = \frac{15}{4}\), or \(y = 3.75\).

Therefore, the y-intercept is located at \(0, 3.75\). Visually, this means that the graph of the equation will cross the y-axis at this point. The y-intercept helps you understand where a line starts from vertically on the graph. Always verify by substituting back into the original formula to make sure it holds true.
graphing equations
Graphing equations using intercepts is a straightforward method because intercepts provide clear points to plot. Here's how it works:
  • First, calculate both the x-intercept and y-intercept.
  • Next, plot these intercepts on your graph.
  • Finally, draw a straight line through the intercepts.
The intercept method is especially helpful for linear equations. It provides you with two definitive points, which is all you need to accurately draw a linear graph.

Once you have these points plotted at \((-3, 0)\) and \(0, 3.75\), use a ruler to connect them. The resulting line represents the graph of \(-5x + 4y = 15\). Double-check by ensuring the line appears straight and passes through both intercepts accurately.

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Most popular questions from this chapter

Find the slope of a line perpendicular to the line passing through the given two points. See Example \(9 .\) \((-1,2)\) and \((-3,6)\)

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((-2,4)\) and \((6,-7)\) \((-6,4)\) and \((5,12)\)

On a quiz, a student was asked to find the slope of the graph of \(y=2 x+3 .\) She answered: \(m=2 x .\) Her instructor marked it wrong. Explain why the answer is incorrect.

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