Problem 29 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (2,-3) \text { and }(-3,7) $$

Step-by-Step Solution

Answer

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

Step 1: Calculate the slope

First, calculate the slope (m) using the formula m = (y2 - y1)/(x2 - x1). By inputting the given points, we yield m = (7 - (-3))/(-3 - 2) = (-10/5), which simplifies to m = -2.

Step 2: Find the y-intercept

Next, use the slope (m) and one of the given points to find the y-intercept (b). For this case, using the point (2,-3) is preferable. The formula is b = y - mx. Therefore, b = -3 - (-2*2) = -3 + 4 = 1.

Step 3: Write the equation

Finally, substitute the values of m and b into the equation y = mx + b to get the equation of the line in slope-intercept form. The equation is therefore y = -2x + 1.

Key Concepts

Calculating Slope

Understanding how to calculate the slope is fundamental when dealing with linear equations. The slope represents how steep a line is and the direction in which the line moves. To calculate the slope, we take two points on a line, \( (x_1, y_1) \) and \( (x_2, y_2) \), and use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For our exercise, we have the points \( (2, -3) \) and \( (-3, 7) \). Plugging these into our formula gives us a slope \( m = \frac{7 - (-3)}{-3 - 2} = \frac{10}{-5} = -2 \).

A positive slope indicates a line rising from left to right, while a negative slope, as in our example, means the line is falling from left to right. It's crucial to grasp this concept, as the slope is a key component in defining the behavior of a linear equation on a graph.

y-intercept

The y-intercept is where a line crosses the y-axis. It's an essential element in the slope-intercept form, which is \( y = mx + b \), where \( b \) is the y-intercept. To find the y-intercept from a graph, look at where the line crosses the y-axis. In equation form, you can calculate it using the slope and a known point on the line. The formula is \( b = y - mx \).

For our example, using the slope \( m = -2 \) and the point \( (2, -3) \), we get \( b = -3 - (-2 \times 2) = -3 + 4 = 1 \). Thus, the y-intercept is 1, meaning our line crosses the y-axis at the point \( (0, 1) \). Knowing the y-intercept is crucial for graphing the line and writing the equation in slope-intercept form.

Linear Equations

A linear equation describes a straight line on a graph and can be presented in various forms, including slope-intercept form. The standard slope-intercept form is \( y = mx + b \) where \( m \) is the slope, and \( b \) is the y-intercept. It's a powerful format because it tells you the slope of the line and where it intercepts the y-axis straight away.

Incorporating the values we found earlier, \( m = -2 \) and \( b = 1 \) gives us the equation \( y = -2x + 1 \). Recognizing how a linear equation represents a line in a coordinate plane helps us to predict patterns, understand relationships, and solve everyday problems.

Graphing Lines

Graphing is a visual way to represent a linear equation. When you graph a line using the slope-intercept form, you start by marking the y-intercept on the y-axis. From there, you use the slope to find another point. A slope of -2 means you move down 2 units and to the right 1 unit from the y-intercept to plot a second point.

Once you have two points, you simply draw a line through them, extending it across the graph. The two points for the example are \( (2, -3) \) and the y-intercept \( (0, 1) \). Connect these, and you have the graph of the equation \( y = -2x + 1 \). Understanding this process is essential in subjects ranging from calculus to real-life applications like economics and engineering.