Problem 22 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x-y=2 \\ 2 x-y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is (4, 2). The lines intersect at (4, 2).

Step 1: Rewrite the equations

Rewrite both equations in slope-intercept form (y = mx + by = mx + by = mx + by = mx + b):\(x - y = 2\) becomes \(y = x - 2\)and \(2x - y = 6\) becomes \(y = 2x - 6\).

Step 2: Graph the first equation

Graph the line represented by \(y = x - 2\): Start by plotting the y-intercept (0, -2). Then use the slope, which is 1 (rise/run), to plot another point, moving up 1 unit and right 1 unit from the y-intercept.

Step 3: Graph the second equation

Graph the line represented by \(y = 2x - 6\): Start by plotting the y-intercept (0, -6). Then use the slope, which is 2, to plot another point, moving up 2 units and right 1 unit from the y-intercept.

Step 4: Identify the point of intersection

Look at the graph to determine where the two lines intersect. The coordinates of this point are the solution to the system of equations.

Step 5: Verify the solution

Check your solution by substituting the coordinates of the intersection point back into the original equations to ensure they satisfy both equations.

Key Concepts

linear equations

Linear equations are mathematical statements that show the relationship between two variables using a straight line when graphed. The general form of a linear equation is \(Ax + By = C\), where\( A\),\( B\), and\( C\) are constants. Linear equations can be written in various forms, the most common being the slope-intercept form, which makes it easier to graph them. Each linear equation represents a straight line, and when you have a system of linear equations, you are looking at multiple lines and their relationships on a graph.

graphing

Graphing is a visual way to represent equations and inequalities on a coordinate plane. For linear equations, graphing involves plotting points that satisfy the equation and then drawing a straight line through these points. In this exercise, we graphed lines to find the solution to a system of linear equations.
To start graphing, you first need the equation in slope-intercept form (\(y = mx + b\)), which allows you to easily identify the y-intercept and the slope.
From there:

• Plot the y-intercept on the y-axis.
• Use the slope (rise/run) to find another point on the line.
• Draw a straight line through these points.

This helps to see where lines intersect, revealing solutions to systems of equations.

slope-intercept form

The slope-intercept form of a linear equation is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
The slope is the rate at which y changes as x changes, often described as 'rise over run.'
The y-intercept is where the line crosses the y-axis.
When you convert equations to slope-intercept form, it simplifies the graphing process. For example, the equation \(x - y = 2\) is rearranged to \(y = x - 2\), making it clear that the y-intercept is -2 and the slope is 1. Using the slope, you move up one unit on the y-axis for every unit you move to the right on the x-axis.

point of intersection

The point of intersection is where the graphs of two equations meet. It represents the solution to a system of equations because that point satisfies both equations.
To find this point by graphing:

• Graph each equation on the same coordinate plane.
• Identify where the lines cross.

The coordinates (x, y) of this intersection point are the values that solve both equations, as verified by substituting them back into the original equations.
In our problem, after graphing \(y = x - 2\) and \(y = 2x - 6\), we found their intersection at a specific point. Verifying this solution ensures accuracy and deepens understanding of solving systems visually.