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Chapter 3: Problem 71

Find all intercepts for each line. Some of these lines have only one intercept. $$y=5 x$$

Short Answer

Expert verified
Both x-intercept and y-intercept are at (0, 0).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. The equation is: \[ y = 5x \]Set y=0:\[ 0 = 5x \]Divide both sides by 5:\[ x = 0 \]So, the x-intercept is at (0, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. The equation is: \[ y = 5x \]Set x=0:\[ y = 5(0) \]\[ y = 0 \]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.

x-intercept
The x-intercept is the point where a line crosses the x-axis. To find this point, you need to set the y-value of the equation to zero. In our example, we have the equation \[y = 5x\].
By setting \(y = 0\), the equation becomes: \[0 = 5x\]
Next, to solve for \(x\), divide both sides by 5:
\(x = 0\).
So, the x-intercept is at (0, 0). This means that the line intersects the x-axis at the origin of the coordinate system. Knowing how to find the x-intercept is important because it helps you understand where the line crosses one of the main axes in a graph. This point can be used to draw and analyze the graph of the equation.
Remember:
  • Set \( y = 0 \) in the equation.
  • Solve for \( x \).
y-intercept
The y-intercept is the point where a line crosses the y-axis. To find this point, set the x-value of the equation to zero. For the equation
\[y = 5x\],
Set \(x = 0\), so the equation becomes:
\[ y = 5(0) \]\[ y = 0\]
Thus, the y-intercept is also at (0, 0). This point tells us where the line meets the y-axis, which is useful for drawing a graph of the function. Just like the x-intercept, knowing the y-intercept helps in understanding the behavior of the line.
Remember:
  • Set x to 0 in the equation.
  • Solve for y.
solving linear equations
Solving linear equations involves finding the values of the variables that make the equation true. In a linear equation such as
\[ y = 5x \],
you can find both x-intercepts and y-intercepts using simple algebraic techniques. To solve for the x-intercept, set y to 0 and solve for x. Similarly, to solve for the y-intercept, set x to 0 and solve for y.
General steps for solving linear equations include:
  • Isolate the variable you need to solve for.
  • Perform operations on both sides of the equation to simplify it.
  • Check your solution by plugging the values back into the original equation.
By understanding these methods, you can solve any linear equation and find important points like intercepts, which help in graphing and interpreting linear relationships.

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Most popular questions from this chapter

Determine which of the ordered pairs \((1,3),(-2,5)\) \((-6,-4),\) and \((7,-8)\) satisfy each compound or absolute value inequality. $$y>5 x \text { and } y>-x$$

Find an equation of the line that goes through the given point and has the given slope. Give the answer in slope-intercept form. See Example 5 (-9, -4) with slope 0

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ h(-3) $$

Determine whether each pair of lines is parallel, perpendicular, or neither. $$2 x-4 y=9, \frac{1}{3} x=\frac{2}{3} y-8$$

Find all intercepts for each line. Some of these lines have only one intercept. $$3 x-5 y=15$$
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