Problem 7 - Algebra 1: Concepts and Skills

Question

find the slope and y-intercept of the equation. $$y+x=15$$

Step-by-Step Solution

Answer

The slope of the equation is -1 and the y-intercept is 15.

Step 1: Isolate y in the equation

In order to isolate y in the equation \(y+x=15\), you need to subtract x from each side of the equation. This gives you the equation in the form \(y= -x + 15\).

Step 2: Identify the slope

In the equation \(y= -x + 15\), the coefficient of x is the slope. Here, our coefficient is -1. So, the slope is -1.

Step 3: Identify the y-intercept

In the equation \(y= -x + 15\), the constant term is the y-intercept. Here, our constant term is 15. So, the y-intercept is 15.

Key Concepts

Slope

In linear equations, the slope is a crucial concept that helps us understand how steep a line is. When given an equation of a line in the form of \(y = mx + b\), the "m" represents the slope. It explains how much the "y" value changes for every change in "x".
To find the slope of an equation, you should first ensure the equation is in the proper form. This is often referred to as the "slope-intercept" form. Once the equation is rearranged appropriately, you just need to look at the coefficient of "x".

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, like in our problem where it is -1, the line falls from left to right.
• A slope of zero means the line is horizontal, with no rise or fall.

The slope gives us important insights about the direction and steepness of a line on a graph, making it easier to visualize relationships between variables.

Y-intercept

The y-intercept is another fundamental element of linear equations. In the equation of a line \(y = mx + b\), the "b" represents the y-intercept. This is the point where the line crosses the y-axis on a graph.
Let's break it down a bit more:

• The y-intercept is the y-coordinate of the point where the line intersects the y-axis.
• This happens when the value of "x" is zero.

For our example equation \(y = -x + 15\), the y-intercept is 15. This means if you were to plot the equation on a graph, the line would intersect the y-axis at the point (0, 15).
Understanding the y-intercept allows us to easily start drawing the line on a graph, providing a fixed point to work from.

Isolate Variable

Isolating a variable is an essential skill in solving equations and is frequently used to rearrange equations to the slope-intercept form. The goal is to have one of the variables by itself on one side of the equation. This simplifies the equation and makes it easier to interpret.
In the given problem, the equation is \(y + x = 15\). To isolate "y", you must:

• Subtract "x" from both sides.
• This results in \(y = -x + 15\), which is now in the slope-intercept form.

By isolating "y", we converted the equation to a form where the slope and y-intercept are easily identifiable. This step is crucial in making the equation more understandable and usable in analyzing the relationship between the variables.