Problem 42 - Algebra 1: Concepts and Skills

Question

Write the equation in slope-intercept form. Then graph the equation. $$2 x+2 y-4=x+5$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -\frac{1}{2}x + \frac{9}{2}\). After graphing this equation, a straight line with a down slope moving from the y-intercept at (0, 4.5) is formed.

Step 1: Simplify the Given Equation

Start this problem by simplifying the given equation. Do so by subtracting \(x\) from both sides of the equation, which simplifies it to: \(x + 2y - 4 = 5\)

Step 2: Write in Slope-Intercept Form

Next, transform the equation into slope-intercept form. Subtract 4 from each side, and divide everything by 2 to simplify. This results in: \(y = -\frac{1}{2}x + \frac{9}{2}\)

Step 3: Interpret the Equation

The equation is now in slope-intercept form, where the slope \(m = -\frac{1}{2}\) and the y-intercept \(b = \frac{9}{2}\) (or 4.5 when converted to decimal form).

Step 4: Graph the Equation

Finally, graph the equation. First plot the point (0, 4.5) on the y-axis. This is the y-intercept. From here, use the slope to determine the next point. Since the slope is -0.5 (or -1/2), move one unit down and two units to the right. Repeat this to plot several points and then connect them to form a straight line, which represents the equation.

Key Concepts

Solving Linear Equations

Solving linear equations involves expressing them in a form that is easy to interpret, typically to find the value of the dependent variable. Linear equations, like the given example, often have variables on both sides. The first step is to simplify the equation, which you can do by performing operations such as addition, subtraction, multiplication, or division to both sides. This helps in isolating the variable of interest.
For example, if we start with an equation like

• \(2x + 2y - 4 = x + 5\)

Subtract \(x\) from both sides, to begin simplifying on the left and eventually isolate \(y\). Make sure whatever operation you perform on one side is also done to the other.
This way, the equation transforms step by step until you reach a simplified form. It is crucial to keep the equation balanced during these operations to maintain equality. Understanding these principles makes it easier to manage any linear equation.

Graphing Linear Equations

Graphing linear equations allows you to visually interpret the relationship between variables. After transforming the equation into the slope-intercept form, the next task is to graph it on a coordinate plane.

• Start by identifying the y-intercept, which in our case is the point where the line crosses the y-axis.
• Using the slope, which tells you how steep the line is, plot additional points starting from the y-intercept.

For example, if the slope is \(-\frac{1}{2}\), it means for every 2 units you move horizontally to the right, you move 1 unit down vertically. A slope of zero would result in a horizontal line, and a positive slope would rise left to right.
This visual representation helps in understanding how changes in one variable affect the other, providing a clear picture of their relationship.

Y-Intercept

The y-intercept of a linear equation is an important concept as it indicates where the line crosses the y-axis. It is represented by \(b\) in the slope-intercept form equation

• \(y = mx + b\)

To find the y-intercept, simply set \(x = 0\) in the equation and solve for \(y\). In our simplified equation, the y-intercept is \(b = \frac{9}{2}\) or \(4.5\).This point is significant as it gives you a starting place for graphing the line. From the y-intercept, you can utilize the slope to find additional points to draw the line accurately.Understanding how to quickly identify and use the y-intercept streamlines the process of graphing linear equations.

Slope Concept

The concept of the slope is central to understanding linear equations and graphs. The slope, denoted as \(m\), measures the rise over run, or how much the line rises vertically for a given horizontal movement.

• If the slope is positive, the line inclines upwards, while a negative slope means the line declines downwards.
• A slope of zero indicates a horizontal line, showing no rise or fall.

In the given example, the slope is \(-\frac{1}{2}\), which means you descend one unit vertically for every two units you move to the right horizontally.
Understanding slope is crucial because it not only tells you about the steepness of the line but also allows you to predict how one variable changes when the other does. It explains the rate of change between the variables, thus offering insights into trends and relationships within the data.