Question

Show graphically that the linear equations \( x - y = 8 \), \( 3x - 3y = 16 \) are inconsistent, i.e. it has no solution.

Hint:

If two linear equations have the same slope but different y-intercepts, they represent parallel lines, meaning the system has no solution.

Answer 1

Step 1: Write the given equations in slope-intercept form.}
We are given the two equations: 1. \( x - y = 8 \) 2. \( 3x - 3y = 16 \) We will first rewrite both equations in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Step 2: Rearrange the first equation.}
For the first equation \( x - y = 8 \): \[ x - y = 8 \] \[ y = x - 8 \] Thus, the first equation becomes \( y = x - 8 \), which has a slope of \( 1 \) and a y-intercept of \( -8 \).
Step 3: Rearrange the second equation.}
For the second equation \( 3x - 3y = 16 \): \[ 3x - 3y = 16 \] \[ 3y = 3x - 16 \] \[ y = x - \frac{16}{3} \] Thus, the second equation becomes \( y = x - \frac{16}{3} \), which has a slope of \( 1 \) and a y-intercept of \( -\frac{16}{3} \).
Step 4: Plot the equations.}
Both equations have the same slope \( 1 \), meaning they are parallel lines. Since the y-intercepts are different (\( -8 \) and \( -\frac{16}{3} \)), the two lines are parallel and will never intersect.

Step 5: Conclusion.}
Since the lines are parallel and do not intersect, the system of equations has no solution. Therefore, the given system of linear equations is inconsistent. % Final Answer Final Answer:
The given system of linear equations has no solution as the lines are parallel and do not intersect.