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Chapter 3: Problem 28

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 4 x-5 y=0 $$

Short Answer

Expert verified
The x-intercept is at (0, 0) and the y-intercept is at (0, 0).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Starting with the equation 4x - 5y = 0 Substitute y = 0: 4x - 5(0) = 0 That simplifies to: 4x = 0 Dividing both sides by 4 gives: x = 0 So, the x-intercept is at (0, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Starting with the equation 4x - 5y = 0 Substitute x = 0: 4(0) - 5y = 0 That simplifies to: -5y = 0 Dividing both sides by -5 gives: y = 0 So, the y-intercept is at (0, 0).

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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 is the point where a graph crosses the x-axis. This happens when the value of y is zero. To find the x-intercept of any equation, we set y to 0 and solve for x. Let's apply this to the equation given in the exercise:
  • Start with equation: 4x - 5y = 0
  • Set y to 0: 4x - 5(0) = 0
  • This simplifies to: 4x = 0
  • Divide both sides by 4: x = 0
So, the x-intercept for the equation is at the point (0, 0). This is the point where the graph crosses the x-axis.
Grasping the y-intercept
The y-intercept is the point where a graph crosses the y-axis. This occurs when the value of x is zero. To find the y-intercept, we set x to 0 and solve for y. Let's see this in action with the provided equation:
  • Start with the equation: 4x - 5y = 0
  • Set x to 0: 4(0) - 5y = 0
  • This simplifies to: -5y = 0
  • Divide both sides by -5: y = 0
Thus, the y-intercept for this equation is also at the point (0, 0). This is the point where the graph crosses the y-axis.
Solving linear equations
Solving linear equations involves finding the values of variables that satisfy the equation. A linear equation is typically in the form ax + by = c. To find the x- and y-intercepts, we solve the equation by substituting the appropriate values.
Here's a step-by-step guide:
  • For the x-intercept, set y to 0 and solve for x.
  • For the y-intercept, set x to 0 and solve for y.
Applying these steps to the example equation 4x - 5y = 0, we found both intercepts to be at (0, 0). This tells us the graph passes through the origin.
Understanding how to find intercepts helps in graphing linear equations and interpreting the relationships between variables.

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Most popular questions from this chapter

Graph each linear equation. $$ y=2 x-5 $$

Complete the given ordered-pair solutions for each equation. Then graph each equation by plotting the points and drawing a line through them. \( \begin{array}{l} y=\frac{2}{3} x+1 \\ \end{array} \) \((0, _____),(3, ____),(-3,_____)\)

\(x=6\)

Write an equation of the line passing through the given point and having the given slope. Give the final answer in slope-intercept form. \((6,-3), m=1\)

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Passes through (2,3) parallel to \(4 x-y=-2\)
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