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Chapter 11: Problem 304

In the following exercises, find the intercepts. $$ x+y=5 $$

Short Answer

Expert verified
The x-intercept is \((5, 0)\) and the y-intercept is \((0, 5)\).

Step by step solution

01

Identify the equation

The given equation is a linear equation: \[x + y = 5\]
02

Find the x-intercept

To find the x-intercept, set \(y = 0\) and solve for \(x\): \[x + 0 = 5\] \[x = 5\] So, the x-intercept is \((5, 0)\).
03

Find the y-intercept

To find the y-intercept, set \(x = 0\) and solve for \(y\): \[0 + y = 5\] \[y = 5\] So, the y-intercept is \((0, 5)\).

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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 line is the point where the line crosses the x-axis. At this point, the value of y is always zero.

To find the x-intercept of a linear equation, such as the one given, you follow these steps:

  • Set y to 0 in the equation.
  • Solve the resulting equation for x.


For the equation \( x + y = 5 \), setting y to 0 gives us \( x + 0 = 5 \), which simplifies to \( x = 5 \).

This means the line crosses the x-axis at the point (5, 0). So, the x-intercept is (5, 0).
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. At this point, the value of x is always zero.

To find the y-intercept of a linear equation, such as the one given, you follow these steps:

  • Set x to 0 in the equation.
  • Solve the resulting equation for y.


For the equation \( x + y = 5 \), setting x to 0 gives us \( 0 + y = 5 \), which simplifies to \( y = 5 \).

This means the line crosses the y-axis at the point (0, 5). So, the y-intercept is (0, 5).
solving equations
Solving equations is a fundamental skill in algebra and mathematics. To solve an equation means to find the values of the variables that make the equation true.

Here's a brief guide to solving linear equations:

  • Isolate the variable: Try to get the variable you are solving for by itself on one side of the equation.
  • Use inverse operations: If the variable is added, subtract it. If it is multiplied, divide it, and so on.
  • Check your solution: Substitute the solution back into the original equation to verify it is correct.


In the given exercise, for finding intercepts, we isolated the variables by setting either x or y to zero and then solved for the remaining variable.

For example, to find the x-intercept, we set y to 0 and solved \( x = 5 \). Similarly, to find the y-intercept, we set x to 0 and solved \( y = 5 \).

By mastering these steps, you can solve a wide range of linear equations and understand the intersections of lines on a graph.

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Most popular questions from this chapter

In the following exercises, identify the most convenient method to graph each line. $$ y=\frac{1}{2} x+2 $$

In the following exercises, find the ordered pairs that are solutions to the given equation. \(5 x+y=10\) (a) \((5,1)\) (b) \((2,0)\) (c) \((4,-10)\)

In the following exercises, graph the line given a point and the slope. $$ (1,5) ; m=-3 $$

How do you find the \(x\) -intercept of the graph of \(3 x-2 y=6 ?\)

In the following exercises, identify the most convenient method to graph each line. $$ y=\frac{2}{3} x-1 $$
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