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Chapter 5: Problem 20

Write an equation of the line satisfying the given conditions. Passing through \((-1,4)\) and \((2,-2)\)

Short Answer

Expert verified
The equation of the line is \(y = -2x + 2\).

Step by step solution

01

Calculate the Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((x_1, y_1) = (-1, 4)\) and \((x_2, y_2) = (2, -2)\), substituting these into the slope formula gives \[ m = \frac{-2 - 4}{2 - (-1)} = \frac{-6}{3} = -2 \]
02

Use Point-Slope Form

The equation of the line can be written using the point-slope form \[ y - y_1 = m(x - x_1) \] Taking one of the points \((x_1, y_1) = (-1, 4)\) and the slope \(m = -2\), plug in these values to get \[ y - 4 = -2(x + 1) \]
03

Simplify to Slope-Intercept Form

Now, simplify the equation to get it into the slope-intercept form \(y = mx + b\). Expand and simplify the equation from Step 2: \[ y - 4 = -2x - 2 \] Adding 4 to both sides, we get \[ y = -2x + 2 \]
04

Verify with the Second Point

To ensure the equation is correct, substitute the second point \((2, -2)\) into the equation \(y = -2x + 2\): \[ -2 = -2(2) + 2 = -4 + 2 = -2 \] Since both sides of the equation are equal, the equation is verified.

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Key Concepts

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

Slope Calculation
Understanding how to calculate the slope of a line is crucial. The slope indicates how steep the line is, and whether it ascends or descends as you move from left to right. To find the slope between two points, use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula represents the change in the y-coordinates (which is the rise) divided by the change in the x-coordinates (the run). In our example, we use the points \((-1, 4)\) and \((2, -2)\).
Plugging these values into the formula: \(
m = \frac{-2 - 4}{2 - (-1)} = \frac{-6}{3} = -2\)
Therefore, the slope \(m= -2\). This tells us the line falls 2 units for every 1 unit it moves to the right.
Point-Slope Form
The point-slope form of a line's equation is a valuable format for writing the equation of a line when you know one point on the line and the slope. The form is described as:
\[ y - y_1 = m(x - x_1) \] where \(m\) is the slope, and \( (x_1, y_1) \) is a specific point on the line. In the exercise, using the point \((-1, 4)\) and the slope \(-2\), we substitute these values into the formula:
\(
y - 4 = -2(x + 1)\)
This step allows us to fill in the known data to create the line equation in a simpler form before further simplifications.
Slope-Intercept Form
The slope-intercept form is probably the most commonly used and recognized form of a linear equation. It is represented as:
\[ y = mx + b \] In this format, \(m\) is the slope, and \(b\) is the y-intercept (the point where the line crosses the y-axis). After rearranging and simplifying the point-slope equation \(y - 4 = -2(x + 1)\), we get:
\(
y - 4 = -2x - 2\)
\( y = -2x + 2 \) Here, \(-2\) is the slope and \(2\) is the y-intercept. This form is easy to interpret and use for graphing lines, as you can immediately identify how steep the line is and where it crosses the y-axis.

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Most popular questions from this chapter

Suppose that the monthly cost of maintaining a car is linearly related to the number of miles driven during that month, and that for a particular model, on average, it costs 160 dollar per month to maintain a car driven 540 miles and 230 dollar per month to maintain a car driven 900 miles. (a) Write an equation relating the cost \(c\) of monthly maintenance and the number \(n\) of miles driven that month. (b) What would be the average cost of maintaining a car driven 750 miles per month? (c) On the basis of this relationship, if your average monthly maintenance is 250 dollar how many miles do you drive per month? (d) Sketch a graph of this equation using the horizontal axis for \(n\) and the vertical axis for \(c\) (e) What is the \(c\) -intercept of this graph? How would you interpret the fact that even if no miles are driven there is still a monthly maintenance cost?

Sketch the graph of the line satisfying the given conditions. Passing through \((1,-2)\) with slope \(\frac{-3}{2}\)

Sets of values are given for variables having a linear relationship. In each case, write the slope-intercept form for the equation of the line corresponding to the given set of values and answer the accompanying question. $$\begin{array}{|l|c|c|} \hline x \text { (Number of hours practicing video game) } & 2 & 3 \\ \hline y \text { (Grade on math exam) } & 75 & 70 \\ \hline \end{array}$$ What would the grade be if a student practices video games for 4 hours?

Solve the following problem algebraically. Be sure to label what the variable represents. Tamika leaves point \(A\) at 10: 00 A.M. traveling due east at 60 kph. One-half hour later, Ramon leaves the same location traveling due west at \(70 \mathrm{kph}\). At what time will they be \(257.5 \mathrm{km}\) apart?

Sketch the graph of \(u-4 v=8\) using the horizontal axis for \(u\) values and the vertical axis for \(v\) values.
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