Questions: Solve the following system of equations by graphing. Then determine whether the system is consistent or inconsistent and whether the equations are dependent or independent. If the system is consistent, give the solution.
7x + 7y = -21
2x + 2y = 16
Graph the linear equations by writing the equations in slope-intercept form:
y = x +
y = x +
Determine whether the system is consistent or inconsistent.
Transcript text: Solve the following system of equations by graphing. Then determine whether the system is consistent or inconsistent and whether the equations are dependent or independent. If the system is consistent, give the solution.
\[
\left\{\begin{array}{l}
7 x+7 y=-21 \\
2 x+2 y=16
\end{array}\right.
\]
Graph the linear equations by writing the equations in slope-intercept form:
\[
\begin{array}{l}
y=\square x+\square \\
y=\square x+\square
\end{array}
\]
Determine whether the system is consistent or inconsistent.
Solution
Solution Steps
Step 1: Convert the equations to slope-intercept form
The given system of equations is:
\[
\begin{cases}
7x + 7y = -21 \\
2x + 2y = 16
\end{cases}
\]
First, convert each equation to the slope-intercept form \(y = mx + b\).
For the first equation:
\[ 7x + 7y = -21 \]
\[ 7y = -7x - 21 \]
\[ y = -x - 3 \]
For the second equation:
\[ 2x + 2y = 16 \]
\[ 2y = -2x + 16 \]
\[ y = -x + 8 \]
Step 2: Graph the equations
Graph the equations \(y = -x - 3\) and \(y = -x + 8\) on the coordinate plane.
Step 3: Determine the consistency and dependency
Since both equations have the same slope (-1) but different y-intercepts (-3 and 8), the lines are parallel and do not intersect.
Final Answer
The system is inconsistent and the equations are independent. There is no solution.