Solve each system of equations using any of the following methods: substitution,
elimination, echelon, or Gauss-Jordan. If there are no solutions, say so. If there are an
infinite number of solutions, parametrize the answer using y as the parameter.
(a) x − 4y = −15
−2x + y = −19
First, solve the second equation for y:
-2x + y = -19
y = 2x - 19
Then substitute this expression for y into the first equation:
x - 4(2x-19) = -15
Simplify and solve for x:
x - 8x + 76 = -15
-7x = -91
x = 13
Now use this value of x to find y:
y = 2x - 19 = 2(13) - 19 = 7
Therefore, the solution to the system is (x,y) = (13,7).
To solve the system of equations:
Step 1: We can start by using the Substitution method. Let's solve one equation for one variable and substitute it into the other equation.
From the second equation, we can express y in terms of x:
-2x + y = -19
=> y = 2x - 19
Step 2: Now substitute this value of y into the first equation:
x - 4y = -15
=> x - 4(2x - 19) = -15
=> x - 8x + 76 = -15
=> -7x = -91
=> x = 13
Step 3: Substitute the value of x back into the second equation to find y:
y = 2x - 19
=> y = 2(13) - 19
=> y = 26 - 19
=> y = 7
So, the solution to the given system of equations is x = 13 and y = 7.