Solve the system of equations.

Question

−4x + 6y = 38
x − 5y = −20

Answer 1

To solve the system of equations, we can use the method of substitution or elimination.

Method 1: Substitution

We can solve the second equation for x and substitute it into the first equation:

x - 5y = -20
x = 5y - 20

Substituting x in the first equation:

-4x + 6y = 38
-4(5y - 20) + 6y = 38
-20y + 80 + 6y = 38
-14y + 80 = 38
-14y = -42
y = 3

Substituting y in the second equation:

x - 5(3) = -20
x - 15 = -20
x = -5

Therefore, the solution to the system of equations is x = -5 and y = 3.

Method 2: Elimination

Multiply the first equation by 5 and the second equation by -4 to eliminate x:

-20x + 30y = 190
-4x + 20y = 80

Adding these two equations together:

-20x + 30y + (-4x + 20y) = 190 + 80
-20x - 4x + 30y + 20y = 270
-24x + 50y = 270

Now we have a new equation:

-24x + 50y = 270
x - 5y = -20

Multiply the second equation by 24 to match the x coefficient with the first equation:

24x - 120y = -480

Adding these two equations together:

-24x + 50y + (24x - 120y) = 270 + (-480)
-70y = -210
y = 3

Substituting y in the second equation:

x - 5(3) = -20
x - 15 = -20
x = -5

Therefore, the solution to the system of equations is x = -5 and y = 3.

Answer 2

How many solution(s) does the system of equations have?
−5x + y = −18
10x −2y = 36

Answer 3

To determine the number of solutions the system of equations has, we can use the method of elimination or substitution.

Method 1: Elimination

Multiply the first equation by 2 and the second equation by 5 to eliminate y:

-10x + 2y = -36
10x - 2y = 36

Adding these two equations together:

-10x + 2y + (10x - 2y) = -36 + 36
0 = 0

Since 0 = 0, we have infinitely many solutions. This means that the system of equations represents equations of the same line, so they intersect at infinitely many points.

Therefore, the system of equations has infinitely many solutions.