/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Write the slope-intercept equati... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 59

Write the slope-intercept equation for the line containing the given pair of points. $$ (1,5) \text { and }(4,2) $$

Short Answer

Expert verified
The equation is \( y = -x + 6 \).

Step by step solution

01

Find the Slope (m)

Use the formula for the slope: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the given points \((1,5)\) and \((4,2)\), we get: \[ m = \frac{2 - 5}{4 - 1} = \frac{-3}{3} = -1 \]
02

Use the Slope-Intercept Form

The slope-intercept form of a line is: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept. We already have \(m = -1\).
03

Find the Y-Intercept (b)

To find \(b\), substitute one of the points (let's use \((1, 5)\)) and the slope \(m = -1\) into the equation \(y = mx + b\): \[ 5 = -1 \cdot 1 + b \] Simplify to find \(b\): \[ 5 = -1 + b \] \[ b = 6 \]
04

Write the Final Equation

Now that we have the slope \(m = -1\) and the y-intercept \(b = 6\), we can write the equation of the line: \[ y = -1x + 6 \] Which simplifies to: \[ y = -x + 6 \]

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Key Concepts

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

Finding the Slope
When we talk about the slope of a line, we are essentially referring to how steep the line is. The slope (often represented as \(m\)) is a measure of the change in the y-coordinate compared to the change in the x-coordinate between two points on the line. More formally, the slope is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This represents the 'rise' over the 'run'. In our specific problem, we are given the points \((1, 5)\) and \((4, 2)\). Plugging these coordinates into the formula, we have: \[ m = \frac{2 - 5}{4 - 1} = \frac{-3}{3} = -1 \] This tells us that our line decreases by 1 unit in the y-direction for every 1 unit increase in the x-direction.
Equation of a Line
The equation of a line in slope-intercept form is a simple yet powerful tool in algebra. The general form is \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) represents the y-intercept, which is where the line crosses the y-axis. By using this form, we can intuitively understand the behavior of the line. Given that we already found the slope to be \( m = -1 \), we can start building our equation. So far, it looks like\: \[ y = -1x + b \] To complete the equation, we need to determine the value of \( b \), the y-intercept.
Y-Intercept
Finding the y-intercept \( b \) is straightforward if we know the slope and a point on the line. We can substitute the slope and one of our points into the equation of the line \( y = mx + b \). Let's use the point \( (1, 5) \): \[ 5 = -1 \times 1 + b \] Solving for \( b \), we get: \[ 5 = -1 + b \ b = 5 + 1 \ b = 6 \] Now that we have evaluated \( b \), we can write the final equation of the line: \[ y = -x + 6 \] So, the line passes through the points given, with a slope of -1 and crosses the y-axis at 6.

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Most popular questions from this chapter

Show that the slope of the line given by \(y=m x+b\) is \(m .\) (Hint: Substitute both 0 and 1 for \(x\) to find two pairs of coordinates. Then use the formula, Slope \(=\) change in \(y /\) change in \(x .\) )

Water freezes at \(32^{\circ}\) Fahrenheit and at \(0^{\circ}\) Celsius. Water boils at \(212^{\circ} \mathrm{F}\) and at \(100^{\circ} \mathrm{C} .\) What Celsius temperature corresponds to a room temperature of \(70^{\circ} \mathrm{F} ?\)

To prepare for Section \(3.8,\) review evaluating algebraic expressions (Section \(1.8)\) $$ 2-x, \text { for } x=-3 $$

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