Problem 52 - Algebra 1

Question

Write the point-slope form of the equation of the line that passes through the two points. $$ (-2,3),(2,-5) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-2,3) and (2,-5) is \( y = -2x + 7 \) in point-slope form.

Step 1: Calculate the Slope

The slope of a line passing through the points \((-2,3)\) and \((2,-5)\) is given by the formula \( m = \frac{{y2 - y1}}{{x2 - x1}} \). Substituting the given points into this formula gives: \(m = \frac{{-5 - 3}}{{2 - (-2)}} = \frac{{-8}}{{4}} = -2 \). So, the slope of the line is -2.

Step 2: Substitute into the Point-Slope Formula

The point-slope form of a line is given by \( y - y1 = m(x - x1) \). Substituting the slope -2 found in Step 1, and one of the points, say (-2,3), into this formula gives: \( y - 3 = -2(x - (-2)) \). Simplifying this gives \( y - 3 = -2x - 4 \).

Step 3: Rewrite in Standard Form

The equation \( y - 3 = -2x - 4 \) can be rewritten in standard form by first distributing the -2 through the parentheses, which gives \( y - 3 = -2x + 4 \). Then, isolating y in the equation results in \( y = -2x + 7 \).

Key Concepts

Slope Calculation

In mathematics, calculating the slope is vital when understanding lines and their relationships on a graph. The slope of a line is essentially a measure of its steepness or incline, indicating how much the line rises or falls as you move from left to right. When provided with two points, such as

• \((-2, 3)\) and
• \((2, -5)\),

the slope is calculated using the formula \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of the given points.
By substituting the values into the equation, we find that the slope \( m = \frac{{-5 - 3}}{{2 - (-2)}} = \frac{{-8}}{{4}} = -2 \).
A negative slope, such as \(-2\), indicates the line descends from left to right.

Linear Equations

Linear equations form the backbone of algebra and are equations that graph as straight lines. These equations are usually represented in various forms, including the point-slope form, slope-intercept form, and standard form.
The point-slope form, given by \( y - y_1 = m(x - x_1) \), is particularly useful when you know a point on the line and the slope.
In our case, substituting the slope \(-2\) and the point

• \((-2, 3)\)

into the point-slope form results in the equation \( y - 3 = -2(x + 2) \).
This equation describes our line in terms of its slope and a specific point, giving valuable insights into the line's direction and position.

Standard Form

The standard form of a linear equation is another way to express a line. It is typically written as \( Ax + By = C \), where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative.
Converting an equation from point-slope or slope-intercept form to standard form can make it easier to understand a graph and to see where a line might intersect the axes.
For the equation \( y - 3 = -2(x + 2) \), we simplify and rearrange it to \( y = -2x + 7 \).
To convert this to standard form, we rearrange terms to isolate the variables on one side, resulting in \( 2x + y = 7 \).
Notice how \( A = 2 \), \( B = 1 \), and \( C = 7 \), which helps in seeing the line's characteristics, such as intercept points on the axes.

• The x-intercept occurs when \( y = 0 \),
• and the y-intercept occurs when \( x = 0 \).

This form is particularly beneficial for easily understanding where lines meet each axis on a graph.