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Chapter 11: Problem 151

In the following exercises, find the intercepts. $$ y=3 x $$

Short Answer

Expert verified
The intercepts are at \((0, 0)\).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). For \(y = 3x\), substitute \(x = 0\): \[ y = 3(0) = 0 \]. So, the y-intercept is at \((0, 0)\).
02

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). For \(y = 3x\), substitute \(y = 0\): \[ 0 = 3x \]. Solving for \(x\), we get \( x = 0 \). So, the x-intercept is at \((0, 0)\).

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Key Concepts

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

intercepts
Intercepts are the points where a graph crosses the axes. These points help us understand the behavior of the graph.

There are two types of intercepts:
  • The y-intercept is where the graph crosses the y-axis.
  • The x-intercept is where the graph crosses the x-axis.

To find the y-intercept of a linear equation like \(y = 3x\), set \(x = 0\) and solve for \(y\). For example, substituting \(x = 0\) in \(y = 3x\) gives us \(y = 0\). So, the y-intercept is at \((0, 0)\).

To find the x-intercept, set \(y = 0\) and solve for \(x\). For the equation \(y = 3x\), substituting \(y = 0\) results in \(0 = 3x\). Solving this, we find \(x = 0\). Thus, the x-intercept is also at \((0, 0)\).

For linear equations, these intercept points provide a clear starting reference for graphing.
graphing linear equations
Graphing linear equations involves plotting lines on a coordinate plane. Start by identifying crucial points like intercepts.

For instance, to graph the equation \(y = 3x\), follow these steps:
  • Find the intercepts. In this case, both the x and y intercepts are at \((0, 0)\).
  • Choose another point on the line. For this example, if \(x = 1\), then \(y = 3 \times 1 = 3\). This gives us the point \((1, 3)\).
  • Plot these points on a graph.
  • Draw a straight line through the points, extending both directions.

Linear equations like \(y = 3x\) produce straight lines, which simplifies the graphing process. Just connect the intercepts and any extra points you calculated.
solving for variables
Solving for variables means finding the unknown values that satisfy an equation.

In the context of linear equations like \(y = 3x\):
  • To solve for \(y\), you need the value of \(x\). If \(x = 2\), for example, then \(y = 3 \times 2 = 6\).
  • To solve for \(x\), you need the value of \(y\). If \(y = 9\), you solve \(3x = 9\), giving \(x = 3\).

This process of substitution makes solving linear equations straightforward. By plugging known values into the equation, you can find the variables鈥� corresponding values. These solutions help in graphing the equations and understanding how variables interact. Remember to perform inverse operations carefully to solve for the unknowns correctly.

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Most popular questions from this chapter

In the following exercises, graph the line given a point and the slope. $$ (-3,4) ; m=-\frac{1}{3} $$

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

In the following exercises, graph the vertical or horizontal lines. $$ x=3 $$

In the following exercises, graph by plotting points. $$ 5 x+y=6 $$

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