Problem 51 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the point and has the given slope. $$ (8,4), m=-5 $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the point (8,4) with a slope of -5 is \(y = -5x + 44\).

Step 1: Use the given point and slope

In the equation \(y = mx + b\), substitute x and y with the values from the given point, and m with the slope. This will look like \(4 = -5 \cdot 8 + b\).

Step 2: Solve for b

After making the substitutions in step one, solve for 'b'. Start by multiplying to get: \(4 = -40 + b\). Then add 40 to both sides of the equation to solve for 'b': \(b = 44\).

Step 3: Write the final equation

Now that we have our slope and y-intercept, we can write the equation in slope-intercept form: \(y = -5x + 44\).

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and describe a straight line when graphed on a coordinate plane. A linear equation typically takes the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form is known as the slope-intercept form. It is not only simple to use but also provides immediate insight into the characteristics of the line.

Understanding linear equations is crucial as they appear frequently in various fields such as physics, economics, and everyday life. They are used to model constant rate changes, making them a powerful tool for solving real-world problems.

• The slope \(m\) represents the rate of change – how much \(y\) changes for a unit change in \(x\).
• The y-intercept \(b\) shows where the line crosses the y-axis, depicting the value of \(y\) when \(x\) is zero.

Knowing how to manipulate and interpret linear equations facilitates the ability to predict and calculate other points on the line, provided you know its rate of change and starting point.

Slope

The slope of a line, denoted by \(m\), is a measure of its steepness. It signifies how much the line rises or falls as you move from left to right horizontally across the graph. Slope is calculated as the ratio of the change in \(y\) (the rise) to the change in \(x\) (the run) between two distinct points on the line.

In the given exercise, the slope is \(m = -5\). This value tells us a few important things:

• A negative slope means the line is descending as you move to the right.
• The higher the absolute value, the steeper the descent. A slope of \(-5\) means that for each step to the right on the \(x\)-axis, the line drops 5 units on the \(y\)-axis.

Understanding the slope is helpful because it allows us to graph linear functions quickly and comprehend their behavior. It is also practical in situations that require analyzing rates, such as speed or cost changes over time.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is represented by the value \(b\) in the slope-intercept form \(y = mx + b\). At this point, the value of \(x\) is zero, making the y-intercept an easy spot to find on the graph. In this exercise, we calculate the y-intercept as \(b = 44\), meaning that if you start at the origin where \(x\) is zero, the line intersects the y-axis at \(y = 44\).

The y-intercept is significant because:

• It provides a starting value for \(y\) when graphing the equation.
• It helps in understanding the context of real-world problems, giving an initial condition or starting point.

By knowing the y-intercept, predictions can be made about the relationship between \(x\) and \(y\) values, supporting decision-making in various practical applications.