Problem 53 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the point and has the given slope. $$ (-8,7), m=2 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form of the line that passes through the given point (-8,7) and has the slope \(m = 2\) is \(y = 2x + 23\).

Step 1: Apply point and slope to the point-slope form

The point-slope equation of the line is given as: \(y - y_1 = m(x - x_1)\). Plug the point (-8,7) where \(x_1 = -8\) and \(y_1 = 7\), and the slope \(m = 2\) into the equation. Hence, the equation becomes: \(y - 7 = 2(x - -8) \).

Step 2: Simplify the equation

Simplify \(x - -8\) to \(x + 8\) and distribute the slope value \(m = 2\) to the terms inside the parentheses. Therefore, the equation becomes: \(y - 7 = 2(x + 8)\) which simplifies to: \(y - 7 = 2x + 16\)

Step 3: Transform into slope-intercept form

To change the equation into slope-intercept form \(y = mx + b\), add 7 to both sides of the equation. The equation becomes: \(y = 2x + 16 + 7\)

Step 4: Final Simplification

Add the constant terms on the right side of the equation, that is 16 + 7 = 23. Thus, the final equation becomes: \(y = 2x + 23\) which represents in the form \(y = mx + b\).

Key Concepts

Point-Slope Form

The point-slope form is a handy way to describe a straight line when you know one point on the line and the slope. The equation for the point-slope form is given by \( y - y_1 = m(x - x_1) \). Here:

• \( (x_1, y_1) \) represents the known point on the line.
• \( m \) is the slope of the line.

When you plug these values into the equation, you can describe any straight line. Let's say you have a point (-8, 7) and a slope of 2 as given in the exercise. By substituting these into the formula, it becomes \( y - 7 = 2(x + 8) \). This form is particularly useful when you want to quickly get to the slope-intercept form, which is often more preferred in final answers.

Linear Equations

Linear equations graph as straight lines and have a constant slope. One of the most prevalent forms of linear equations is the slope-intercept form, represented as \( y = mx + b \). This formula is beneficial because:

• \( m \) is the slope of the line, which shows how steep the line is.
• \( b \) is the y-intercept, where the line crosses the y-axis.

Linear equations are fundamental in algebra; they appear in various fields, including economics, physics, and engineering. Converting a linear equation from the point-slope to slope-intercept form is common practice, as it provides a clearer view of the slope and y-intercept. From the exercise, \( y - 7 = 2(x + 8) \) is converted to \( y = 2x + 23 \). Here, the converted equation shows that the slope \( m \) is 2, and the y-intercept \( b \) is 23.

Slope

The slope of a line refers to how steep the line is and is commonly represented by the letter \( m \). It is a measure of the change in the vertical direction (rise) relative to a change in the horizontal direction (run). The formula for slope is given by \( m = \frac{\Delta y}{\Delta x} \).

• 'Delta' denotes change, so \( \Delta y \) is change in y and \( \Delta x \) is change in x.
• If the slope is positive, the line ascends from left to right.
• If the slope is negative, the line descends from left to right.

Understanding the slope is crucial as it helps determine the direction and steepness of the line. In the exercise example, the given slope was 2, indicating that for every one unit increase in x, y increases by two units. This concept is crucial for interpreting and graphing linear equations accurately.