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Chapter 3: Problem 36

Find the intercepts and graph them. $$ -25 x+3 y=75 $$

Short Answer

Expert verified
The x-intercept is \((-3, 0)\), and the y-intercept is \((0, 25)\).

Step by step solution

01

Identify the equation type

The given equation is \(-25x + 3y = 75\), which is a linear equation in standard form \(Ax + By = C\).
02

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). \[-25x + 3(0) = 75 \] \[-25x = 75\] Divide both sides by \(-25\): \[x = \frac{75}{-25} = -3\]So, the x-intercept is \((-3, 0)\).
03

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). \[-25(0) + 3y = 75\] \[3y = 75\] Divide both sides by \(3\):\[y = \frac{75}{3} = 25\]So, the y-intercept is \((0, 25)\).
04

Plot the intercepts on a graph

On a coordinate plane, plot the x-intercept \((-3, 0)\), which is 3 units to the left on the x-axis, and the y-intercept \((0, 25)\), which is 25 units up on the y-axis.
05

Draw the line

Draw a line passing through both intercepts \((-3, 0)\) and \((0, 25)\) on the graph. This line represents the equation \(-25x + 3y = 75\).

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Key Concepts

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

Understanding the X-Intercept
The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. At this point, the value of y is always 0. To find the x-intercept of the given equation \(-25x + 3y = 75\), we substitute \(y = 0\) and solve for \(x\). This gives us:
  • \(-25x + 3(0) = 75\)
  • Solving \(-25x = 75\) results in \(x = -3\)
Therefore, the x-intercept is \((-3, 0)\). This means the graph will cross the x-axis at this point. Finding the x-intercept helps determine one of the two starting points needed to graph the equation.
Exploring the Y-Intercept
The y-intercept is where the graph crosses the y-axis, and at this point, the value of x is always 0. To find the y-intercept for the equation \(-25x + 3y = 75\), set \(x = 0\) and solve for \(y\):
  • \(-25(0) + 3y = 75\)
  • Solving \(3y = 75\) gives \(y = 25\)
So, the y-intercept is \((0, 25)\). This result provides the second essential point for drawing the graph. With the y-intercept, you know exactly where the line crosses the y-axis.
Graphing Linear Equations
Graphing a linear equation involves plotting points and drawing a line that represents the equation. For the equation \(-25x + 3y = 75\), you have calculated two intercepts: the x-intercept \((-3, 0)\) and the y-intercept \((0, 25)\). Here's how you graph it:
  • Begin by plotting the x-intercept \((-3, 0)\) on the coordinate plane. This point is three units to the left on the x-axis.
  • Next, plot the y-intercept \((0, 25)\). This point is 25 units up on the y-axis.
  • Draw a straight line passing through both intercepts. This line is the graphical representation of the equation.
By using both intercepts, you ensure the line is accurately represented. Graphing linear equations using intercepts is a straightforward way to visualize relationships in mathematics.

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