Given the equations:
[tex]\begin{gathered} (1)3x-4y=10 \\ (2)y=\frac{3}{4}x+3 \end{gathered}[/tex]To solve the system, follow the steps below.
Step 01: Write the equations in the slope-intercept form.
An equation in the slope-intercept form is: y = mx + b, where m is the slope and b is the y-intercept.
So, let's write the first equation using this form. To do it, let's subtract 3x from both sides.
[tex]\begin{gathered} 3x-4y-3x=10-3x \\ -4y=-3x+10 \end{gathered}[/tex]Now, let's divide both sides by -4.
[tex]\begin{gathered} \frac{-4}{-4}y=\frac{-3x+10}{-4} \\ y=\frac{3}{4}x-\frac{10}{4} \end{gathered}[/tex]The second equation is already in the slope-intercept form.
Step 02: Compare both equations.
[tex]\begin{gathered} (1)\frac{3}{4}x-\frac{10}{4} \\ (2)\frac{3}{4}x+3 \end{gathered}[/tex]As can be seen, both have the same slope and different y-intercepts.
It is known that parallel lines have the same slope and different y-intercepts.
So, the lines are parallel.
A system of parallel lines has no solution since the lines do not have an interception point.
Answer: No solution.