Problem 36 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \((-2, 2)\).

Step 1: Write Each Equation in Slope-Intercept Form

The first equation is already almost in slope-intercept form, which is \( y = mx + b \). Rearrange it to solve for \( y \). The first equation becomes: \[ y = -\frac{1}{2}x + 1 \].The second equation is a vertical line, which already is in the form \(x = -2\).

Step 2: Plot the First Equation

Plot the first equation \( y = -\frac{1}{2}x + 1 \). Start by identifying the y-intercept \( b = 1 \). Then, use the slope \( -\frac{1}{2} \): from the y-intercept, go down 1 unit and right 2 units to get another point.

Step 3: Plot the Second Equation

Plot the second equation \( x = -2 \). Since this is a vertical line, draw a straight line through \( x = -2 \) on the graph.

Step 4: Find the Intersection Point

The solution to the system of equations is where the two lines intersect. Locate the intersection point on the graph, which should be at the coordinates \((-2, 2)\).

Step 5: Verify the Solution

Verify the intersection point \((-2, 2)\) by substituting back into the original equations.For \( x + 2y = 2 \):\[ -2 + 2(2) = 2 \]\[ 2 = 2 \] - True.For \( x = -2 \):\[ x = -2 \] - True.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra. It's all about plotting points on a graph to visualize how equations behave.
To graph a linear equation, you need to recognize its structure. Linear equations can come in various forms, but they all represent straight lines when plotted.
Here are key steps:

• Convert the equation to slope-intercept form, if necessary.
• Identify key points such as the y-intercept.
• Use the slope to find additional points.
• Draw the line connecting the points.

Graphing helps you see the relationships between variables and is essential for solving systems of equations.

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the easiest ways to graph it. It is written as:
$$ y = mx + b $$
Here:

• \textslope (m): The rate at which the line rises or falls.
• \textintercept (b): The point where the line crosses the y-axis.

This form is convenient because it immediately shows the slope and y-intercept.
For example, in the equation $$ y = -\frac{1}{2}x + 1 $$, the slope is $$ -\frac{1}{2} $$ and the y-intercept is $$ 1 $$. From the y-intercept, you go down 1 unit and right 2 units to plot the line.

Intersection of Lines

The intersection of two lines on a graph represents the solution to a system of linear equations.
This point is where both equations are simultaneously true.
To find it, you follow these steps:

• Graph each equation on the same set of axes.
• Look for the point where the lines cross.

In our example, the system:
onumber\begin{cases} x + 2y = 2 ewline x = -2 \textend{cases}
Intersect at $$ (-2, 2) $$, meaning both equations hold true at this point.

Verification of Solutions

After finding the intersection point, it's crucial to verify the solution.
This ensures that the point indeed satisfies both original equations.
For our system:
$$ \begin{cases} x + 2y = 2 $ ewline x = -2 \textend{cases}

• Plug $$ x = -2 $$ and $$ y = 2 $$ back into both equations.
• Check that both sides of the equations balance.

This gives:
$$-2 + 2(2) = 2$$
and $$ x = -2$$ which verifies the solution as $$(-2, 2)$$.