Problem 12 - Algebra 1: Concepts and Skills

Question

Rewrite the equation in slope-intercept form. $$3 x+y=-11$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -3x - 11\)

Step 1: Isolate y

Starting with \(3x + y = -11\), subtract \(3x\) from both sides:
\(y = -3x - 11\)

Step 2: Identify slope and y-intercept

The equation \(y = -3x - 11\) is in slope-intercept form \(y = mx + b\) with:

• Slope \(m = -3\)
• \(y\)-intercept \(b = -11\)

Key Concepts

Linear Equations

Understanding linear equations is foundational for algebra and beyond. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in various forms, one of which is the slope-intercept form, where the equation is expressed as y = mx + b. Here, m represents the slope—indicating how steep the line is, and b signifies the y-intercept—where the line crosses the y-axis.

In the context of the given exercise, the equation 3x + y = -11 can be transformed to slope-intercept form. This form is very useful because it gives immediate visual information about the line, namely its steepness and where it intersects the y-axis. It's also an excellent starting point for graphing the line, making predictions, and understanding the relationship between variables.

Remember, in a linear equation, the variables should always appear to the first power. If they're squared or in a different form, then it isn't a linear equation.

Isolating Variables

Isolating variables is a technique often used to solve equations, including linear ones. It involves moving all terms containing the variable of interest to one side of the equation and all other terms to the opposite side. This is achieved by using basic arithmetic operations—addition, subtraction, multiplication, and division—while making sure that whatever you do to one side of the equation, you also do to the other to maintain balance.

In the example problem, we isolated y by subtracting 3x from both sides of the original equation. This is a fundamental step to finding the slope-intercept form of a linear equation, because we want y to be by itself on one side. This principle can be applied to virtually any equation to simplify and solve it for a specific variable. It's also crucial in preparation for graphing, because having y isolated allows us to see how y changes in response to x.

Graphing Lines

Graphing lines on the coordinate plane is a visual way to represent linear equations. Once in slope-intercept form, y = mx + b, a line can easily be graphed by identifying the y-intercept (0, b) and using the slope, m, to find another point. The slope indicates the rise over the run—that is, how many units to move up or down for each unit moved to the right.

For the given equation, once written as y = -3x - 11, we know the line crosses the y-axis at (0, -11), and the slope tells us that for every one unit we move horizontally, we move three units down vertically. This slope, having a negative sign, indicates that the line is falling as it moves from left to right. By plotting these points and drawing a line through them, the equation's meaning becomes clear visually—showing the exact linear relationship between x and y.