Problem 24

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope 1 and $x$ -intercept $(-2,0)$

Step-by-Step Solution

Verified

Answer

The standard form of the equation is $ x - y = -2 $.

1Step 1: Understand the problem

We need to find the equation of a line with a given slope and a specific x-intercept. The slope is 1, and the x-intercept is given as (-2, 0). The final form of the equation should be in standard form.

2Step 2: Write the equation in slope-intercept form

The slope-intercept form of a line is given by $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. Since the slope $ m = 1 $, we have $ y = x + b $. We recognize that at the x-intercept point $(-2,0)$, so we will use this point to find $ b $.

3Step 3: Substitute the x-intercept to find b

Substitute $(x, y) = (-2, 0)$ into the equation $ y = x + b $ to find $ b $. Thus, $ 0 = -2 + b $. Solving for $ b $, we get $ b = 2 $. Now the equation is $ y = x + 2 $.

4Step 4: Convert slope-intercept form to standard form

The standard form of a line equation is $ Ax + By = C $. Starting from $ y = x + 2 $, we rearrange this to get the form: $ x - y = -2 $. To comply with the conventions of standard form, we rearrange to $ x - y = -2 $. Thus, the equation is already in the standard form.

Key Concepts

Slope-Intercept FormX-InterceptStandard Form

Slope-Intercept Form

A linear equation can be conveniently represented using the slope-intercept form. This format is useful because it directly provides two important details about the line:

The slope (m), which indicates how steep the line is.
The y-intercept (b), which is where the line crosses the y-axis.

The general formula is given by:\[ y = mx + b\]In this problem, the slope is given as 1, leading to the initial equation:\[ y = x + b\]To find the value of "b", we use the x-intercept point $(-2, 0)$. By inserting these coordinates, we get a complete equation:

Substitute x = -2 and y = 0 into the equation.
Solve for b: $0 = -2 + b$ which leads to $b = 2$.

So, the complete slope-intercept form for this line is:\[ y = x + 2\]

X-Intercept

The x-intercept of a line is the point where the line crosses the x-axis. This is an important characteristic because it can tell us a lot about the line's behavior.

At the x-intercept, the y-value is always 0.
For the equation of a line in any form, we determine the x-intercept by substituting y = 0 and solving for x.

In this problem, the given x-intercept is $x = -2$ at the point $(-2, 0)$. This piece of information helped us find the y-intercept "b" in the slope-intercept form by substitution.Knowing the x-intercept is crucial as it helps in solving equations and graphing the line, serving as one of the essential points on the line.

Standard Form

Standard form is another way to express a linear equation, and it emphasizes the coefficients of the x and y terms.

The general structure of standard form is $ Ax + By = C $.
It aligns the x and y terms on one side of the equation, leaving a constant on the other side.

To convert from slope-intercept form $ y = x + 2 $ to standard form:

Rearrange to get $ x - y = -2 $, which already fits the format.
Ensure all terms align with the conventions, typically with A, B, and C as integers.

The final standard form of the equation is:\[ x - y = -2\]This form is particularly useful in various applications like analyzing systems of equations and ensuring compatibility with mathematical solutions.

Problem 24

Problem 25

Other exercises in this chapter

Problem 24

Explain how the following functions can be obtained from $y=x^{3}$ by basic transformations: (a) $y=x^{3}+1$ (b) $y=(x+1)^{3}-1$ (c) $y=-3(x-2)^{3}$

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Problem 24

(a) Graph $f(x)=x, x \geq 0$, and $g(x)=x^{2}, x \geq 0$, together, in one coordinate system. (b) For which values of $x$ is $f(x) \geq g(x)$, and for w

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Problem 25

Explain how the following functions can be obtained from $y=1 / x$ by basic transformations: (a) $y=1-\frac{1}{x}$ (b) $y=-\frac{1}{x-1}$ (c) \(y=\frac{x}

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Problem 25

(a) Graph $f(x)=x^{2}$ and $g(x)=x^{3}$ for $x \geq 0$, together, in one coordinate system. (b) Show algebraically that $$ x^{2} \geq x^{3} $$ for \(0 \le

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