Problem 48 - Precalculus

Question

Solve each system of linear equations by graphing. $$\begin{array}{r} x-2 y=-1 \\ -x-y=-5 \end{array}$$

Step-by-Step Solution

Answer

The solution is \( x = 3 \) and \( y = 2 \).

Step 1: Rewrite Equations in Slope-Intercept Form

First, we will rewrite both equations in the slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.Starting with the first equation:\[ x - 2y = -1 \]Subtract \( x \) from both sides:\[ -2y = -x - 1 \]Divide each term by \(-2\):\[ y = \frac{1}{2}x + \frac{1}{2} \]Now, for the second equation:\[ -x - y = -5 \]Add \( x \) to both sides:\[ -y = x - 5 \]Divide each term by \(-1\):\[ y = -x + 5 \]

Step 2: Graph Both Equations

Next, we'll graph each equation on the coordinate plane. For \( y = \frac{1}{2}x + \frac{1}{2} \):- Start at the y-intercept \( (0, \frac{1}{2}) \).- From there, use the slope \( \frac{1}{2} \), which means rise 1 unit and run 2 units.For \( y = -x + 5 \):- Start at the y-intercept \( (0, 5) \).- From there, use the slope \(-1\), which means rise -1 unit for every 1 unit run.

Step 3: Identify the Point of Intersection

With the lines plotted, observe where they intersect. The intersection represents the solution to the system of equations. On the graph, we see the lines intersect at the point \((3, 2)\). This means that the solution to the system of equations is \( x = 3 \) and \( y = 2 \).

Step 4: Verify the Solution

To ensure the solution is correct, substitute \( x = 3 \) and \( y = 2 \) back into the original equations:1. For \( x - 2y = -1 \):\[ 3 - 2(2) = 3 - 4 = -1 \]This holds true.2. For \( -x - y = -5 \):\[ -3 - 2 = -5 \]This also holds true.Both equations are satisfied, confirming the solution is correct.

Key Concepts

Graphing Method

To solve a system of linear equations using the graphing method, the first step is to plot each equation as a graph on a coordinate plane. This involves translating the equations into visual lines. By doing this, we can identify the point where the lines intersect, which represents the solution to the system.
A clear graph allows you to see where two equations meet:

• The horizontal axis (x-axis) and vertical axis (y-axis) form a grid.
• Each equation is represented by a line on this grid.

The point where the lines cross is crucial. It shows where both equations are true at the same time. It's a straightforward yet effective way to visualize and find the solution to the equations involved.

Slope-Intercept Form

The slope-intercept form of an equation is written as: \[ y = mx + b \] Here, \( m \) represents the slope of the line, indicating how steep it is. The \( b \) is the y-intercept, where the line meets the y-axis.
When given equations in standard form, convert them to slope-intercept form:

• Start by moving terms to isolate \( y \) on one side of the equation.
• This makes it easy to graph since you now have a clear slope and y-intercept to use.

Using this form, you can quickly determine the starting point of the line on the graph (via the intercept) and how the line rises or falls (via the slope). This clarity makes graphing efficient and effective.

Point of Intersection

The point of intersection is where two lines on a graph meet. This point represents the solution to the system of equations since it satisfies both equations simultaneously.

How to find it:

• Plot each line using the slope and y-intercept derived from the equations.
• Observe where the lines intersect on the graph.

This intersection point is essential because it shows the values of \( x \) and \( y \) that solve the system. You can verify by plugging these values back into the original equations to check whether they hold true. This step confirms the accuracy of the graphical solution.