Problem 44 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (-8.5,6.75),(3.33,-9.75) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (-8.5,6.75) and (3.33,-9.75) is \(y = mx + b\) where \(m\) and \(b\) are the slope and y-intercept found in previous steps.

Step 1: Calculate the Slope

The slope \(m\) of a line passing through the points \((-8.5,6.75)\) and \((3.33,-9.75)\) is calculated as follows: \(m = \frac{y2 - y1}{x2 - x1} = \frac{-9.75 - 6.75}{3.33 - -8.5}\)

Step 2: Calculate the y-intercept

Then the y-intercept \(b\) can be found using the formula \(b = y - mx\), using one of the given points, for example \((-8.5,6.75)\) and the calculated slope \(m\): \(b = 6.75 - m * -8.5\)

Step 3: Write the Equation

Lastly, substitute the found values of \(m\) and \(b\) into the slope-intercept form of the line equation: \(y = mx + b\)

Key Concepts

Equation of a Line

The equation of a line represents all the points that lie on that line. In this exercise, we're asked to find the equation of a line that passes through two given points: \((-8.5, 6.75)\) and \((3.33, -9.75)\). The most popular form for a line equation in two dimensions is the slope-intercept form. This form is given by the equation:\[y = mx + b\]

• \(m\) represents the slope of the line.
• \(b\) is the y-intercept, or where the line crosses the y-axis.

This equation makes it easy to understand how steep the line is and where it will intersect the y-axis. Understanding the structure of this equation is important since it allows you to quickly sketch a line or understand its behavior from just two numbers. With the slope-intercept form, you only need the slope and y-intercept to determine the complete equation.

Calculate Slope

The slope of a line measures its steepness and direction. It tells you how much the line rises or falls as you move from left to right. The formula for calculating the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Example Calculation

Using the points \((-8.5, 6.75)\) and \((3.33, -9.75)\) from our exercise, we plug them into the formula:\[m = \frac{-9.75 - 6.75}{3.33 - (-8.5)}\]This calculation gives us the slope \(m\). Remember that:

• The slope is positive if the line rises from left to right.
• The slope is negative if the line falls from left to right.
• If the line is horizontal, the slope is zero.

A thorough understanding of how to calculate and interpret the slope is key in analyzing lines and their equations.

Y-Intercept

The y-intercept \(b\) is the point where the line crosses the y-axis on a graph. Knowing the y-intercept is crucial because it tells us where the line begins when \(x = 0\). Using the equation from our exercise, we can find \(b\) once we know the slope \(m\).

Finding the Y-Intercept

Using the formula: \[b = y - mx\]Select one of the points, such as \((-8.5, 6.75)\), and with the slope we've just found, substitute into the formula:\[b = 6.75 - m(-8.5)\]This calculation will give us the value of \(b\).

• The y-intercept is particularly easy to spot on a graph because it's where the line touches the y-axis.
• It provides a starting point for graphing the line when combined with the slope.

Once you know \(b\), you simply add it to the slope-intercept equation with your calculated slope \(m\) to fully define the line.