Problem 18 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x-y=3 \\ 2 x-y=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is x = 1, y = -2.

Step 1 - Write equations in slope-intercept form

Rewrite each equation in the form of y = mx + b. For the first equation, x - y = 3, solve for y: y = x - 3. For the second equation, 2x - y = 4, solve for y: y = 2x - 4.

Step 2 - Graph the first equation

Graph y = x - 3. Start by plotting the y-intercept (0, -3). Then use the slope (which is 1, or 1/1) to plot another point by going up 1 unit and right 1 unit from the intercept.

Step 3 - Graph the second equation

Graph y = 2x - 4. Start by plotting the y-intercept (0, -4). Then use the slope (which is 2, or 2/1) to plot another point by going up 2 units and right 1 unit from the intercept.

Step 4 - Identify the intersection point

Examine the graph to find the point where the two lines intersect. The intersection point is the solution to the system of equations.

Step 5 - Verify the solution

Substitute the coordinates of the intersection point back into the original equations to ensure they satisfy both equations.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a crucial skill in algebra. It helps us visualize solutions to equations and systems of equations. To graph a linear equation, you must first put it in slope-intercept form: \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept. Plot the y-intercept on the graph, then use the slope to find another point. Draw a straight line through these points. Remember to label your axes and plot points accurately to get an accurate graph.

Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \). It's named so because it clearly shows the slope (\( m \)) and the y-intercept (\( b \)). The slope describes how steep the line is and the direction it goes. The y-intercept is where the line crosses the y-axis. To convert an equation to this form, solve for \( y \). For example, for the equation \( x - y = 3 \), rearrange it to get \( y = x - 3 \). Similarly, for \( 2x - y = 4 \), rearrange it to get \( y = 2x - 4 \).

Intersection Point

In a system of linear equations, the intersection point of the graphs of the equations is the solution. This point represents the values of \( x \) and \( y \) that satisfy both equations simultaneously. To find it, graph both equations on the same set of axes and look for the point where they cross. For our example, the lines \( y = x - 3 \) and \( y = 2x - 4 \) intersect at (1, -2). This is seen by plotting both lines carefully according to their slopes and intercepts.

Verifying Solutions

Once you identify the intersection point, you must verify that it indeed solves both equations. To do this, substitute the coordinates of the intersection point back into the original equations. For instance, for the point \((1, -2)\), substitute \( x = 1 \) and \( y = -2 \) into both original equations: 1) \( x - y = 3 \): \( 1 - (-2) = 3 \) which is true. 2) \( 2x - y = 4 \): \( 2(1) - (-2) = 4 \) which is also true. Thus, the solution is verified.