Problem 25 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (2,-3),(-3,7) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (2,-3) and (-3,7) is \(y = -2x + 1\).

Step 1: Calculate the slope

The formula for slope (m) is given by \((y_2-y_1) / (x_2-x_1)\). Here, \(x_1 = 2\), \(y_1 = -3\), \(x_2 = -3\), and \(y_2 = 7\). Substituting these values into the formula gives \(m = (7 - (-3)) / (-3 - 2) = 10 / -5 = -2\).

Step 2: Calculate the y-intercept

Using the slope intercept form \(y = mx + b\), and substituting \(m = -2\), \(x = 2\), and \(y = -3\), the y-intercept (b) is calculated as follows: \(-3 = -2*2 + b\), which simplifies to \(-3 = -4 + b\). Therefore, \(b = 1\).

Step 3: Write the equation

Finally, substituting \(m = -2\) and \(b = 1\) into the slope-intercept equation gives us the equation of the line: \(y = -2x + 1\).

Step 4: Graph the points and the line

Plot the given points (2, -3) and (-3, 7) on the graph and draw a straight line through them. This line will represent the equation \(y = -2x + 1\).

Key Concepts

Graphing Linear Equations

Graphing linear equations is like creating a map that shows the relationship between two variables, typically x and y. When we graph a linear equation, we're essentially drawing a straight line that goes through all the points that meet the equation's criteria. To graph a line, you need only two points, but checking with a third ensures accuracy.

To put it simply, every point on that line is a solution to the equation. You can start by plotting the points on a grid. In our exercise, we have two points, (2, -3) and (-3, 7), which tell us where the line passes. After marking these points on graph paper or using digital graphing tools, draw a straight line that connects them, and extend it across the grid. This visual representation mirrors the equation's behavior and the change in values.

Calculating Slope

The slope of a line measures its steepness and direction. Think of it as how fast a line travels upward or downward as you move from left to right across a graph. Calculating slope is crucial because it tells us the rate at which y changes with respect to x.

To calculate it, we use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(m\) represents the slope, and \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of any two points on the line. With our given points (2, -3) and (-3, 7), we found that the slope \(m = -2\). This negative value means the line goes down as we move from left to right, which would indicate a downhill slope in real life.

Y-intercept

The y-intercept is the point where our line crosses the y-axis. It’s like telling us the starting position of y before x starts to change. To find it, we often rearrange the equation to the slope-intercept form \(y = mx + b\), with \(m\) as the slope and \(b\) as the y-intercept. Knowing the slope, we can plug in our known x and y from a point on our line to discover \(b\).

In the example given, after calculating the slope as -2, we used one of the given points, plugged the values into the slope-intercept form, and solved for \(b\), finding it to be 1. This means our line crosses the y-axis at the point (0, 1). It's an important part of the equation because it gives us a fixed point from which we can draw our line.