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Chapter 3: Problem 22

Find the intercepts and graph them. $$ -2 x+3 y=6 $$

Short Answer

Expert verified
x-intercept: (-3, 0), y-intercept: (0, 2), graph the line through these points.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \(-2x + 3y = 6\). This simplifies to \(-2x = 6\). Solving for \( x \) gives \( x = -3 \). Thus, the x-intercept is \((-3, 0)\).
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \(-2x + 3y = 6\). This simplifies to \(3y = 6\). Solving for \( y \) gives \( y = 2 \). Thus, the y-intercept is \((0, 2)\).
03

Graph the intercepts

On a coordinate plane, plot the points \((-3, 0)\) and \((0, 2)\). Draw a straight line through these points to represent the equation \(-2x + 3y = 6\). This line is the graph of the given equation.

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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 of a linear equation is the point where the graph of the equation crosses the x-axis. This means that at this point, the value of y is equal to zero. To find the x-intercept, we simply need to substitute 0 for y in the equation and solve for x.

For the equation \(-2x + 3y = 6\), substituting y with 0 gives you \(-2x = 6\). Solving this for x, we divide both sides by -2 to get \(x = -3\). So, the x-intercept is the point (-3, 0) on the graph.

Key points to remember about x-intercepts:
  • They always have a y-coordinate of 0.
  • They show where the line meets the x-axis.
Understanding the y-intercept
The y-intercept of a linear equation is the point where the graph crosses the y-axis. Here, the x value is equal to zero. To find the y-intercept, we need to set x to 0 in the equation and solve for y.

Going back to our equation \(-2x + 3y = 6\), if we set x to 0, we have \(3y = 6\). Solving for y involves dividing by 3, resulting in \(y = 2\). Thus, the y-intercept is located at (0, 2) on the graph.

Some highlights regarding y-intercepts:
  • They always have an x-coordinate of 0.
  • They indicate where the line meets the y-axis.
Graphing linear equations using intercepts
Graphing a linear equation is much simpler when you understand how to use intercepts. Intercepts give us two precise points through which we can draw a line. You only need two points to determine a straight line, making intercepts an efficient tool for graphing.

For the equation \(-2x + 3y = 6\), we have already determined the intercepts: (-3, 0) and (0, 2). When you plot these points on a coordinate plane, the x-intercept at (-3, 0) and the y-intercept at (0, 2), you can draw a straight line through them.
  • Start by placing a dot at each intercept on the graph.
  • Draw a line through these two points, extending it in both directions.
  • This line represents all possible solutions to the equation \(-2x + 3y = 6\).
Graphing using intercepts allows you to quickly visualize the relationship between variables in an equation.

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