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Chapter 3: Problem 35

Write the slope-intercept equation for the line with the given slope and containing the given point. $$ m=\frac{7}{4} ;(4,-2) $$

Short Answer

Expert verified
The equation is \( y = \frac{7}{4}x - 9 \).

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute the Given Slope

The problem provides the slope \( m = \frac{7}{4} \). Substitute this value into the slope-intercept form: \( y = \frac{7}{4}x + b \).
03

Use the Given Point

The problem provides a point on the line, \( (4, -2) \). This means when \( x = 4 \), \( y = -2 \). Substitute these values into the equation: \( -2 = \frac{7}{4}(4) + b \).
04

Solve for the Y-Intercept

Simplify the equation to solve for \( b \). First, calculate \( \frac{7}{4}(4) = 7 \), so the equation becomes \( -2 = 7 + b \). To isolate \( b \), subtract 7 from both sides: \( b = -2 - 7 \), which simplifies to \( b = -9 \).
05

Write the Final Equation

Now that we know the slope \( m = \frac{7}{4} \) and the y-intercept \( b = -9 \), we can write the equation of the line: \( y = \frac{7}{4}x - 9 \).

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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 measures how steep it is. It's represented by the letter 'm' in the slope-intercept form of a linear equation: \( y = mx + b \). Think of the slope as the 'rise over run,' which means how much the line goes up (or down) for each unit it goes across. For example, a slope of \( \frac{7}{4} \) means for every 4 units you move to the right, you move 7 units up.
The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Slopes can be positive, negative, zero, or undefined:
  • Positive slope: The line goes up as you move from left to right.
  • Negative slope: The line goes down as you move from left to right.
  • Zero slope: The line is horizontal and does not go up or down.
  • Undefined slope: The line is vertical.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. It is represented by 'b' in the slope-intercept form \( y = mx + b \). For our equation, \( y = \frac{7}{4}x - 9 \), the y-intercept is -9. This means when \( x = 0 \), \( y = -9 \).
Finding the y-intercept in a given problem often involves substituting the slope and a known point\((x_1, y_1)\) back into the linear equation and solving for 'b'. For example, if you have a point (4, -2) and a slope of \( \frac{7}{4} \), you'd substitute these values in:
\( -2 = \frac{7}{4}(4) + b \).
You'd solve the equation step-by-step to find 'b':
  • Simplify to \( -2 = 7 + b \)
  • Subtract 7 from both sides to get \( b = -9 \)
Now, you鈥檝e found your y-intercept!
Linear Equations
Linear equations describe lines on a graph and are typically written in the slope-intercept form \( y = mx + b \). This form makes it easy to see the slope and y-intercept of the line straight away.
To graph a linear equation:
  • Start by plotting the y-intercept \( b \) on the y-axis.
  • Use the slope \( m \) to determine your next point. For example, with a slope of \( \frac{7}{4} \), move 4 units right (run) and 7 units up (rise).
  • Draw a line through these points.
Linear equations are incredibly useful in various fields, from economics to engineering, because they model relationships with a constant rate of change. If you have a specific slope and point, you can quickly form the equation of your line and understand how any changes in \( x \) will affect \( y \).

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