Solve using the elimination method. Show your work. If the system has no solution or an infinite number of solutions, state this.
2x + 10y = 58
-14x + 8y = 140
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.
To solve this system of equations using the elimination method, we need to eliminate one variable from the equations by manipulating them in a way that will result in the same coefficient for that variable. Here's how:
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.