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Chapter 11: Problem 45

Write an equation of the line that passes through the points \((-2,-5)\) and \((1,-1)\)

Short Answer

Expert verified
The equation of the line is \[ y = \frac{4}{3}x - \frac{7}{3} \].

Step by step solution

01

- Find the Slope

The first step to writing the equation of the line is to find the slope. The slope (m) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(\frac{y_2 - y_1}{x_2 - x_1}\). So, for the points \((-2, -5)\) and \(1, -1)\), we calculate as follows: \[ m = \frac{-1 - (-5)}{1 - (-2)} = \frac{-1 + 5}{1 + 2} = \frac{4}{3} \]
02

- Use the Point-Slope Formula

Next, use the point-slope formula to write the equation of the line. The point-slope form is given by \(y - y_1 = m(x - x_1)\). Using the point \((1, -1)\) and the slope \(m = \frac{4}{3}\), we get: \[ y - (-1) = \frac{4}{3}(x - 1) \]
03

- Simplify the Equation

Simplify the equation from Step 2 to write it in slope-intercept form \(y = mx + b\). \[ y + 1 = \frac{4}{3}(x - 1) \] Distribute the \( \frac{4}{3} \): \[ y + 1 = \frac{4}{3}x - \frac{4}{3} \] Subtract 1 from both sides: \[ y = \frac{4}{3}x - \frac{4}{3} - 1 \] Simplify the constants: \[ y = \frac{4}{3}x - \frac{7}{3} \]

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Key Concepts

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

Slope
The slope of a line is a measure of its steepness or inclination. It's denoted by the letter m. A positive slope means the line rises from left to right, while a negative slope means it falls.
The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For our points \((-2, -5)\) and \((1, -1)\), plug in the values:
\[- m = \frac{-1 - (-5)}{1 - (-2)} = \frac{-1 + 5}{1 + 2} = \frac{4}{3} \] This tells us the line rises 4 units for every 3 units it runs horizontally.
Point-Slope Formula
The point-slope formula is useful when you know one point on the line and the slope. It's expressed as: \[ y - y_1 = m(x - x_1) \]
This formula essentially says that the difference in the y-coordinates between any point \(x, y\) on the line and the known point \(x_1, y_1\) is proportional to the difference in their x-coordinates, scaled by the slope, m.
Using our point \((1, -1)\) and slope \((m = \frac{4}{3})\), substitute into the formula:
\[ y - (-1) = \frac{4}{3}(x - 1) \]
Simplifying further, you'll see:
\[ y + 1 = \frac{4}{3}(x - 1) \] This equation is a starting point to help us formulate other forms of the line equation.
Slope-Intercept Form
The slope-intercept form of a linear equation is probably the most familiar. It's written as \[ y = mx + b \]
Here, m is the slope and b is the y-intercept - the point where the line crosses the y-axis.
We begin with the point-slope form and rearrange it into slope-intercept form.
From: \[ y + 1 = \frac{4}{3}(x - 1) \] Distribute the slope:
\[ y + 1 = \frac{4}{3}x - \frac{4}{3} \] Isolate y by subtracting 1 from both sides:
\[ y = \frac{4}{3}x - \frac{4}{3} - 1 \]
Combine the constants:
\[ y = \frac{4}{3}x - \frac{7}{3} \]
Now, the line's equation is in slope-intercept form, showing the relationship between y and x directly.

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Most popular questions from this chapter

In Exercises \(85-94\), solve the equations using the square root method. Round off your answers to the nearest hundredth. $$3 k^{2}=20$$

In Exercises \(65-74\), solve the given equation. For quadratic equations, choose either the factoring method or the square root method, whichever you think is the easier to use. $$4(x+1)=\frac{9}{x+1}$$

In Exercises \(85-94\), solve the equations using the square root method. Round off your answers to the nearest hundredth. $$4 x^{2}=19.7$$

Use the method you think is the most appropriate to solve the given equation. Check your answers by using a different method. $$\frac{x+2}{x}=\frac{5}{x-4}$$

In Exercises \(75-84\), round your answer to the nearest tenth where necessary. One leg of a right triangle is 10 in., and the hypotenuse is 26 in. Find the length of the other leg.
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