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Chapter 1: Problem 12

The graph of each equation is a straight line. Graph the equation after finding the \(x\)-and the \(y\) -intercepts. (since you are given that the graph is a line, you need only plot two points before drawing the line.) $$x=2 y-4$$

Short Answer

Expert verified
The x-intercept is (-4, 0) and the y-intercept is (0, 2).

Step by step solution

01

Rewrite the Equation in Standard Form

Start by rewriting the equation in the form of a standard line equation, which is typically written as \(Ax + By = C\). For the given equation \(x = 2y - 4\), subtract \(2y\) from both sides to get \(x - 2y = -4\).
02

Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) in the equation \(x - 2y = -4\). When \(y = 0\), the equation becomes \(x = -4\). Thus, the \(x\)-intercept is \((-4, 0)\).
03

Find the y-intercept

To find the \(y\)-intercept, set \(x = 0\) in the equation \(x - 2y = -4\). Solving for \(y\) when \(x = 0\) gives \(-2y = -4\); dividing both sides by -2, we get \(y = 2\). Therefore, the \(y\)-intercept is \((0, 2)\).
04

Plot and Draw the Line

Plot the points for the \(x\)-intercept \((-4, 0)\) and the \(y\)-intercept \((0, 2)\) on the Cartesian plane. Draw a straight line through these points extending in both directions. This line represents the graph of the equation \(x = 2y - 4\).

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Key Concepts

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

x-intercept
When graphing linear equations, finding the x-intercept is a fundamental step. The x-intercept is the point where the graph of an equation crosses the x-axis. At this point, the value of y is always zero. This means to find the x-intercept, you substitute 0 in for y in the equation and solve for x.
For example, in the equation \(x - 2y = -4\), setting \(y = 0\) transforms it to \(x = -4\). Thus, the x-intercept is at the point \((-4, 0)\).
  • To find the x-intercept of any linear equation, set y to zero and solve for x.
  • The x-intercept is the point \( (x, 0) \).
y-intercept
The y-intercept is another essential element for graphing linear equations. It is the point where the line crosses the y-axis, which means the x-coordinate at this point is always zero. To find the y-intercept, simply set x to 0 in your equation and solve for y.
Consider our equation \(x - 2y = -4\). By setting \(x = 0\), the equation becomes \(-2y = -4\). Solving for y, you divide both sides by -2, giving \(y = 2\). Therefore, the y-intercept is at \((0, 2)\).
  • To find the y-intercept, set x to zero and solve for y.
  • The y-intercept is the point \( (0, y) \).
Standard Form of a Linear Equation
The standard form of a linear equation is a way of writing the equation so it is easily recognizable and workable. A linear equation in standard form looks like \(Ax + By = C\), where A, B, and C are integers, and A should be non-negative. This format is useful because it allows for easy identification of intercepts and graphing.
In our exercise, the original equation was \(x = 2y - 4\). To convert it to standard form, we moved the terms around to rearrange the expression into \(x - 2y = -4\). Here, A is 1, B is -2, and C is -4.
  • Standard form is useful for quick identification of intercepts.
  • The coefficients A, B, and C should ideally be integers to simplify calculations.

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Most popular questions from this chapter

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x|>0$$

Explain why there are no real numbers that satisfy the equation \(\left|x^{2}+4 x\right|=-12\).

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x|<2$$

Find an equation for the line passing through the two given points. Write your answer in the form \(y=m x+b\). (a) (7,9) and (-11,9) (b) \((5 / 4,2)\) and \((3 / 4,3)\) (c) (12,13) and (13,12)

Verify the identity $$\left(y_{2}-y_{1}\right) /\left(x_{2}-x_{1}\right)=\left(y_{1}-y_{2}\right) /\left(x_{1}-x_{2}\right)$$ What does this identity tell you about calculating slope?
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