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Chapter 11: Problem 137

In the following exercises, find the intercepts. $$ 3 x+y=6 $$

Short Answer

Expert verified
The x-intercept is (2, 0) and the y-intercept is (0, 6).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y=0in the equation and solve for x.Starting with the given equation: $$3x + y = 6$$Setting y = 0we get:$$3x + 0 = 6$$Simplified as:$$3x = 6$$Now solve for x:$$x = \frac{6}{3}$$Therefore, the x-intercept is:$$x = 2$$So the coordinates for the x-intercept are:(2, 0).
02

Find the y-intercept

To find the y-intercept, set x=0in the equation and solve for y.Starting with the given equation: $$3x + y = 6$$Setting x = 0we get:$$3(0) + y = 6$$Simplified as:$$y = 6$$Therefore, the y-intercept is:$$y = 6$$So the coordinates for the y-intercept are:(0, 6).

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Key Concepts

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

X-Intercept
In a linear equation, the x-intercept is the point where the graph of the equation crosses the x-axis. This means the y-coordinate at this point is zero. To find the x-intercept, we set y to 0 and solve for x.

Using the given example, let's find the x-intercept:

Start with the equation: \[3x + y = 6\]

Set y to 0:

\[3x + 0 = 6\]

Thus, the equation simplifies to:

\[3x = 6\]

Divide both sides by 3 to isolate x:

\[x = \frac{6}{3}\]

Simplified, x equals 2. Therefore, the x-intercept is at the point (2, 0).
Y-Intercept
The y-intercept in a linear equation is the point where the graph crosses the y-axis. At this point, the x-coordinate is zero. To find the y-intercept, we set x to 0 and solve for y.

Using our example, we'll find the y-intercept:

Start with the equation:

\[3x + y = 6\]

Set x to 0:

\[3(0) + y = 6\]

The equation simplifies to:

\[y = 6\]

So, when x is 0, y equals 6. Therefore, the y-intercept is at the point (0, 6).
Linear Equations
Linear equations are equations of the first order and can be graphically represented as straight lines. The general form of a linear equation in two variables, x and y, is:

\[Ax + By = C\]

In this form, A, B, and C are constants. In our example:

\[3x + y = 6\]

A linear equation can have one or more intercepts. The intercepts are extremely useful as they give us specific points to draw the line on a graph. For example, in our equation, we found:

  • The x-intercept, where y = 0, to be (2, 0)
  • The y-intercept, where x = 0, to be (0, 6)
These intercepts allow us to graph the line accurately.

Remember, finding intercepts helps in understanding the behavior of linear equations and plotting them correctly on a graph.

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