Question

how many solutions does the system of equations below have?
x - 6y = -10
3x - 18y = -20
no solution
one solution
infinitely many solutions
submit

Explanation:

Step1: Rewrite equations in slope-intercept form

For the first equation \( x - 6y = -10 \), solve for \( y \):
\( -6y = -x - 10 \)
\( y=\frac{1}{6}x+\frac{10}{6}=\frac{1}{6}x+\frac{5}{3} \)

For the second equation \( 3x - 18y = -20 \), solve for \( y \):
\( -18y=-3x - 20 \)
\( y=\frac{-3x}{-18}+\frac{-20}{-18}=\frac{1}{6}x+\frac{10}{9} \)

Step2: Analyze slopes and y-intercepts

The slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
For the first equation, slope \( m_1=\frac{1}{6} \), y-intercept \( b_1 = \frac{5}{3} \).
For the second equation, slope \( m_2=\frac{1}{6} \), y-intercept \( b_2=\frac{10}{9} \).

Since \( m_1 = m_2 \) (same slopes) but \( b_1
eq b_2 \) (different y-intercepts), the lines are parallel and never intersect. So the system has no solution.

Answer:

no solution