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Chapter 3: Problem 40

In the following exercises, find the intercepts for each equation.. $$ x-2 y=8 $$

Short Answer

Expert verified
The x-intercept is (8, 0), and the y-intercept is (0, -4).

Step by step solution

01

Identify the x-intercept

To find the x-intercept, set y to 0 and solve the equation for x. Start with the equation: x - 2y = 8 Set y = 0: x - 2(0) = 8 This simplifies to: x = 8 So, the x-intercept is (8, 0).
02

Identify the y-intercept

To find the y-intercept, set x to 0 and solve the equation for y. Start with the equation: x - 2y = 8 Set x = 0: 0 - 2y = 8 Simplify and solve for y: -2y = 8 Divide both sides by -2: y = -4 So, the y-intercept is (0, -4).

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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 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, you simply set y to 0 in the equation and solve for x.

For example, in the equation \( x - 2y = 8 \), to find the x-intercept, we set y to 0:
\( x - 2(0) = 8 \)
This simplifies to:
\( x = 8 \)
So, the x-intercept is at the point (8, 0).

This means that when the graph crosses the x-axis, the x-value at that point is 8.
y-intercept
The y-intercept of a linear equation is where the graph intersects the y-axis. At this point, the value of x is always 0. To find the y-intercept, you set x to 0 and solve for y in the equation.

Using the same example, \( x - 2y = 8 \), we set x to 0:
\( 0 - 2y = 8 \)
This simplifies to:
\( -2y = 8 \)
Now, divide both sides by -2:
\( y = -4 \)
So, the y-intercept is at the point (0, -4).

This means that when the graph crosses the y-axis, the y-value at that point is -4.
linear equations
Linear equations form straight lines when graphed on a coordinate plane. They can typically be written in the form \( ax + by = c \), known as the standard form, or \( y = mx + b \), known as the slope-intercept form.

In our example, \( x - 2y = 8 \) is in standard form.

These equations have constant rates of change and no exponents or variables multiplied together.

The graph of a linear equation has no curves, and the slope (m) is constant throughout. Linear equations are fundamental in algebra as they represent simple relationships between variables.
solving for variables
To solve for variables in an equation, isolate the variable you are solving for on one side of the equation. This often involves using operations like addition, subtraction, multiplication, and division.

For the equation \( x - 2y = 8 \):
  • If we want to solve for x, we set y to 0 and solve for x by performing operations to isolate x on one side.
  • To solve for y, we would isolate y by first setting x to a specific value, like 0, and then solving the resulting equation for y.


This process allows you to find specific values for variables, which can then be used to plot points on a graph or understand relationships within the equation.

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Most popular questions from this chapter

Graph the linear inequality: \(y \leq-\frac{1}{2} x+4\)

In the following exercises, (a) graph each function (b) state its domain and range. Write the domain and range in interval notation. $$ f(x)=3 x^{2} $$

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Explain why, in some graphs of linear inequalities, the boundary line is solid but in other graphs it is dashed.

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