Multiply the first equation by 7, then add the two equations.
78y = 546
Solve for y. Insert that value into the first equation and solve for x. Check by inserting both values into the second equation.
2x + 10y = 58
-14x + 8y = 140
78y = 546
Solve for y. Insert that value into the first equation and solve for x. Check by inserting both values into the second equation.
Step 1: Multiply both sides of the second equation by 2 to make the coefficients of x match:
-14x + 8y = 140 --> -28x + 16y = 280
Step 2: Now, we need to eliminate x. To do this, we'll add the two equations together:
(2x + 10y) + (-28x + 16y) = 58 + 280
Combining like terms:
-26x + 26y = 338
Step 3: Simplify the equation:
-26(x - y) = 338
Divide both sides of the equation by -26:
x - y = -13
So, the simplified form of the system of equations is:
x - y = -13 ...(1)
2x + 10y = 58 ...(2)
Now, let's solve this simplified system using the elimination method:
Step 1: Multiply equation (1) by 2 to make the coefficient of x match the coefficient of x in equation (2):
2(x - y) = -26
2x - 2y = -26
Step 2: Add equation (2) and the simplified equation (1) together:
(2x + 10y) + (2x - 2y) = 58 - 26
4x + 8y = 32
Step 3: Simplify the equation:
4(x + 2y) = 32
Divide both sides of the equation by 4:
x + 2y = 8
So, the simplified form of the system of equations is:
x + 2y = 8 ...(3)
2x + 10y = 58 ...(4)
Now, we have a new system of equations:
x + 2y = 8 ...(3)
2x + 10y = 58 ...(4)
To solve this new system, we can use the substitution method or continue using the elimination method.
From equation (3), we can express x as a function of y:
x = 8 - 2y
Substituting this value of x into equation (4):
2(8 - 2y) + 10y = 58
16 - 4y + 10y = 58
6y = 42
y = 7
Now we can substitute the value of y into equation (3):
x + 2(7) = 8
x + 14 = 8
x = -6
Therefore, the solution to the system of equations is x = -6 and y = 7.
In conclusion, the system has a unique solution, which means there is a single point of intersection between the two lines represented by the equations.