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Chapter 3: Problem 20

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ -3 x+2 y=12 $$

Short Answer

Expert verified
The x-intercept is \((-4, 0)\) and the y-intercept is \((0, 6)\).

Step by step solution

01

Finding the x-intercept

To find the x-intercept, set y to 0 and solve for x. Start with the equation: \(-3x + 2y = 12\) Set \(y = 0\): \(-3x + 2(0) = 12\) So, \(-3x = 12\) Divide both sides by -3: \(x = -4\) Thus, the x-intercept is \((-4, 0)\).
02

Finding the y-intercept

To find the y-intercept, set x to 0 and solve for y. Use the same initial equation: \(-3x + 2y = 12\) Set \(x = 0\): \(-3(0) + 2y = 12\) So, \(2y = 12\) Divide both sides by 2: \(y = 6\) Therefore, the y-intercept is \((0, 6)\).

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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 the graph of a linear equation crosses the x-axis. To find it, we set y to 0 in the equation and solve for x. This is because any point on the x-axis will have a y-value of 0. Let's break down the steps to find the x-intercept for the given equation [-3x + 2y = 12y].
  • Start by setting y = 0. So our equation becomes: [-3x + 2*(0) = 12y=-3x=12=x=-4y].
  • Next, solve this equation by dividing both sides by -3 to get x = -4. This means the x-intercept is at (-4, 0).
The x-intercept gives you valuable information about the graph. It tells you where the line will cross the x-axis, and it helps in plotting the graph accurately.
Understanding the y-intercept
The y-intercept is the point where the graph of a linear equation crosses the y-axis. To determine it, we set x to 0 and solve for y. This is because any point on the y-axis has an x-value of 0. Let's understand how to find the y-intercept using the equation: [-3x + 2y = 12y].
  • Start by setting x = 0. This modifies our equation to: [-3*(0) + 2y = 12y2y=12=62y].
  • Next, solve this equation by dividing both sides by 2 to get y = 6. That means the y-intercept is at (0, 6).
The y-intercept is another crucial piece of information when it comes to graphing a line. Knowing this, along with the x-intercept, allows you to plot the line more precisely.
Understanding linear equations
Linear equations represent straight lines when graphed on a coordinate plane. They are commonly written in the form [ax + by = cy], where a, b, and c are constants. Key characteristics include:
  • These equations have a constant rate of change, known as the slope.
  • They can have one, none, or infinite solutions.
  • The general form is useful for finding intercepts, as shown in the exercises.
Here's how to think about a linear equation like [-3x + 2y = 12y]:
  • The terms involving x and y show the relationship between the two variables.
  • The equation balances to a constant value, creating a straight line when plotted.

When finding intercepts or solving these equations, remember:
  • The x-intercept occurs where y=0.
  • The y-intercept occurs where x=0.
Both are essential for graphing and understanding the behavior of linear relationships.

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Most popular questions from this chapter

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