Problem 12 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (-2,5),(-6,-8) $$

Step-by-Step Solution

Answer

The equation of the line that passes through the points (-2,5) and (-6,-8) in slope-intercept form is \( y = 3.25x + 6.5 \)

Step 1: Compute the Slope

The formula for finding the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = (y_2 - y_1) / (x_2 - x_1) \). Plugging in the given points \((-2,5)\) and \((-6,-8)\), the slope becomes \( m = (-8 - 5) / (-6 - (-2)) = -13 / -4 = 3.25 \)

Step 2: Solve for the y-intercept

The slope-intercept form is \( y = mx + b \). Having found the slope, choose one of the points, for example \((-2,5)\), and substitute its values, as well 'm', into the equation to solve for 'b': \( 5 = 3.25*(-2) + b \) which simplifies to \( b = 5 - 3.25*(-2) = 6.5 \)

Step 3: Write the Final Equation

Now that both 'm' and 'b' have been found, you can substitute these into the slope-intercept equation to give the equation of the line: \( y = 3.25x + 6.5 \)

Key Concepts

Linear Equations

Linear equations are fundamental expressions in mathematics that define a straight line when graphed on a coordinate plane. They take the form of an equation with two variables, typically "x" and "y", and show how these variables relate to each other.

A linear equation can generally be written in different forms, but one of the most common is the slope-intercept form:

• The standard form: Ax + By = C, where A, B, and C are constants.
• The slope-intercept form: y = mx + b, where "m" represents the slope of the line and "b" is the y-intercept.

If you remember that these equations graph as a straight line, you can better understand how they work in real-life situations, like predicting expenses or analyzing trends.

Slope

The slope of a line is a measure of how steep the line is. It communicates the rate at which "y" changes concerning "x" as you move along the line. To find the slope between two points - \((x_1, y_1)\) and \((x_2, y_2)\) - use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

For the points (-2, 5) and (-6, -8), using this formula gives us the slope of 3.25.

• A positive slope means the line inclines upward from left to right.
• A negative slope indicates the line declines downward.
• A slope of zero refers to a perfectly horizontal line.

Understanding the slope is crucial because it affects the angle of the line on a graph, and thus, reflects real-world relationships like speed, cost per item, etc.

y-intercept

The y-intercept is the point where a line crosses the y-axis of a graph. It shows the value of "y" when "x" is zero, which is the starting point of the line when no progress along the x-axis is made.

In the slope-intercept form equation, y = mx + b, "b" represents this y-intercept. To find it, once you have the slope, substitute one of the points into the equation along with the slope to solve for "b". In our solution, by choosing the point (-2, 5) with a slope of 3.25, we calculated the y-intercept to be 6.5.

• The y-intercept provides a quick way to describe a line's position in a graph.
• It helps to predict values quickly when analyzing a graph or pattern based on a linear equation.

Without this essential value, it would be impossible to precisely locate where a line begins on the y-axis, making it crucial for graph interpretation and creation.