Problem 50 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} -x+2 y=-2 \\ y=-x-1 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is (0, -1).

Step 1: Rewrite Equations in Slope-Intercept Form

The given equations are already in forms easily translatable to slope-intercept form, which is given by the formula: \(y = mx + b\)First equation: \(-x + 2y = -2\) Rewrite to solve for y: \(2y = x - 2\)Then divide by 2: \(y = \frac{1}{2}x - 1\). Second equation: already in slope-intercept form \(y = -x - 1\).

Step 2: Identify the Slope and Y-Intercept

For \(y = \frac{1}{2}x - 1\): Slope (\(m\)) is \(\frac{1}{2}\) and y-intercept (\(b\)) is -1.For \(y = -x - 1\): Slope (\(m\)) is -1 and y-intercept (\(b\)) is -1.

Step 3: Plot the Y-Intercepts

Start by plotting the y-intercept of each equation on the y-axis.For \(y = \frac{1}{2}x - 1\), plot the point (0, -1).For \(y = -x - 1\), plot the point (0, -1).

Step 4: Plot Additional Points Using the Slope

Use the slope to find another point for each line. For \(y = \frac{1}{2}x - 1\), from (0, -1), move up 1 unit and right 2 units to plot another point (2, 0).For \(y = -x - 1\), from (0, -1), move down 1 unit and right 1 unit to plot another point (1, -2).

Step 5: Draw the Lines

Draw straight lines through the points plotted for each equation.The line for \(y = \frac{1}{2}x - 1\) should pass through points (0, -1) and (2, 0).The line for \(y = -x - 1\) should pass through points (0, -1) and (1, -2).

Step 6: Identify the Intersection

Find the intersection point of the two lines. The intersection point represents the solution to the system of equations. Upon graphing, the intersection appears at the point (0, -1).

Key Concepts

graphing linear equations

Graphing linear equations is a visual way to represent solutions to equations. It helps you see how different equations relate to each other. To graph a linear equation, you first need to write it in a friendly format known as the slope-intercept form: \(y = mx + b\)
Here, \(m\) represents the slope and \(b\) is the y-intercept. Let's break down what these terms mean:

• Slope (\