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Chapter 11: Problem 342

Find the \(x\) -intercept and \(y\) -intercept of the equation \(3 x-y=6\)

Short Answer

Expert verified
The x-intercept is 2 and the y-intercept is -6.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 and solve the equation for x. Starting with the equation: \[ 3x - y = 6 \] Set y = 0: \[ 3x - 0 = 6 \] So, solve for x: \[ 3x = 6 \] \[ x = \frac{6}{3} \] \[ x = 2 \] Thus, the x-intercept is 2.
02

- Find the y-intercept

To find the y-intercept, set x to 0 and solve the equation for y. Starting with the equation: \[ 3x - y = 6 \] Set x = 0: \[ 3(0) - y = 6 \] So, solve for y: \[ -y = 6 \] \[ y = -6 \] Thus, the y-intercept is -6.

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Key Concepts

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

Understanding x-intercepts
To find the x-intercept of an equation, you need to determine where the graph of the equation crosses the x-axis. This happens when the value of y is zero. Think of the x-intercept as the point where a line touches the horizontal axis.
Let's break it down:
  • Start with the given equation
  • Replace y with 0
  • Solve the resulting equation for x
For example, with the equation \(3x - y = 6\), setting y to 0 gives us \(3x - 0 = 6\). After simplifying, you get \(3x = 6\) and then solving for x results in \(x = 2\).
So, the x-intercept is the point (2, 0). This means the line will cross the x-axis at 2.
Remember, whenever you need an x-intercept, just set y to zero and solve for x.
Understanding y-intercepts
To find the y-intercept, you're looking for the point where the graph crosses the y-axis. This occurs when the value of x is zero. The y-intercept represents the point at which a line touches the vertical axis.
These are the steps:
  • Use the original equation
  • Replace x with 0
  • Solve the simplified equation for y
For the equation \(3x - y = 6\), substituting x with 0 results in \(3(0) - y = 6\). This simplifies to \( - y = 6\).
Solve for y and you get \( y = -6\).
Thus, the y-intercept is the point (0, -6). Here, the line will intersect the y-axis at -6.
This quick substitution technique works for any linear equation when finding intercepts.
Solving linear equations
Solving linear equations involves finding the values of variables that satisfy the given equation. Linear equations are typically in the form of \( ax + by = c \), where a, b, and c are constants.
Here are some strategies to tackle them:
  • Identify what you're solving for (usually x or y)
  • Look to isolate that variable by performing operations like addition, subtraction, multiplication, or division
  • Apply the same operation to both sides of the equation to maintain balance
In our example, the equation \( 3x - y = 6 \) was solved by isolating x and y for the intercepts.
To find the x-intercept, we set y to 0 and solved for x. For the y-intercept, we set x to 0 and solved for y.
Understanding these steps allows you to solve any linear equation for its variables and intercepts effectively.

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Most popular questions from this chapter

In the following exercises, graph by plotting points. $$ y=-3 x $$

In the following exercises, graph each equation. $$ y=\frac{1}{4} x-2 $$

Complete the table to find three solutions to the equation \(4 x+y=8\)

Explain how you would choose three \(x\) -values to make a table to graph the line \(y=\frac{1}{5} x-2\)

Do you prefer to graph the equation \(4 x+y=-4\) by plotting points or intercepts? Why?
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