27x = 3y + 78 is the first equation.
subtract 78 from both sides to get 27x - 78 = 3y
solve for y to get y = 27x / 3 - 78/3
simplify to get y = 9x - 26
3y = 90 - 6x
rearrange the terms in descending order of degree tp get 3y = -6x + 90
divide both side by 3 to get y = -6/3 * x + 90/3
simplify to get y = -2x + 30
they are now both in slope intercept form of y = mx + b where m is the slope and b is the y-intercept.
if the slopes are the same and the y-intercepts are the same, the lines are identical and you have an infinite number of possible solutions.
if the slopes are the same and the y-intercepts are different, the lines are parallel and you have no possible solutions.
if the slopes are not the same, the lines are not identical or parallel and you have one possible solution.
these lines do not have the same slope and you therefore have one possible solution.
i graphed both equations and the graph shows one possible solution.
the red line is caused by the first equation after it have been placed in slope intercept form.
the blue line is caused by the second equation after it has been placed in slope intercept form.
the graph shows one solution at the coordinate point of (5.091,19.818)
that means x = 5.091 and y = 19.818 is the one common solution to both equations.
i solved them algebraically and this is what i got.
start with:
27x = 3y + 78
3y = 90 - 6x
solve for 3y in both equations to get:
3y = 27x - 78
3y = -6x + 90
subtract the second equation from the first to get 0 = 33x - 168
add 168 to both sides of the equatino to get 168 = 33x
solve for x to get x = 168 / 33 = 5.090909091
solve for y in either original equation to get y = 19.81818182
round to 3 decimal places and you get x = 5.091 and y = 19.818 as shown on the graph.
here's the graph.
there is only one solution for the system of these two equations.