Question

Use the linear system below. $$\begin{array}{l} y=x+3 \\ y=2 x+3 \end{array}$$ Graph the system. Explain what the graph shows.

Short answer

The graph displays two parallel lines, which indicates there is no solution for the system since these lines never intersect.

Step-by-step solution

- Identifying the equation format

Both equations are already in the form \(y=mx+c\), where \(m\) is the slope of the line and \(c\) is the y-intercept. For the equation \(y=x+3\), \(m=1\) and \(c=3\). For the equation \(y=2x+3\), \(m=2\) and \(c=3\).

- Graphing the lines

Plot the y-intercept first. In both cases, the y-intercept is 3, so plot the point (0,3). Finally, use the slope to determine another point on each line and draw a straight line connecting the points. The slope of the first line is 1, so it can be thought of as the fraction 1/1. Starting from the y-intercept, count up 1 and over 1 and plot a point. Draw a line through these points. Repeat the process for the second line, but this time with a slope of 2. Start from the y-intercept, count up 2 and over 1 and plot a point. Draw a line through these points.

- Interpretating the graph

Examining the graph, it's noticable that the lines are parallel but do not coincide, since they have the same y-intercept but different slopes. This signifies that the system of linear equations has no solution. The effects can be checked by equating the two linear equations, showing the solution either in fraction form or decimal form, \(1 \neq 2\). As it's a false statement, clearly indicating that the system of equations have no solution and lines do not intersect anywhere. Both lines represent infinite many points, but there is no common point for given two lines, which indicates that the system has no solution.