Solve the system of equations using the linear combination method.
{9x+5y=35
2x+5y=0
since 5y appears in both equations, just subtract and you get
7x = 35
x = 5
Then use that in either equation to figure y.
9 x + 5 y = 35
-
2 x + 5 y = 0
__________
9 x - 2 x + 5 y - 5 y = 35 - 0
7 x = 35
x = 35 / 7
x = 5
Sustitute this in 9 x + 5 y = 35
9 ∙ 5 + 5 y = 35
45 + 5 y = 35
5 y = 35 - 45
5 y = - 10
y = - 10 / 5
y = - 2
Step 1: Start by writing down both equations of the system:
Equation 1: 9x + 5y = 35
Equation 2: 2x + 5y = 0
Step 2: Multiply Equation 2 by -9 so that the coefficients of x in both equations will cancel each other out when the two equations are added together.
(-9)(2x + 5y) = (-9)(0)
-18x - 45y = 0
Simplifying Equation 2: -18x - 45y = 0
Step 3: Add Equation 1 and Equation 2 together:
(9x + 5y) + (-18x - 45y) = 35 + 0
(9x + (-18x)) + (5y + (-45y)) = 35
-9x - 40y = 35
Simplifying the sum: -9x - 40y = 35
Step 4: Now we have a new equation, -9x - 40y = 35, which is a result of adding the two original equations. This new equation represents a line that is equivalent to the original system.
Step 5: Solve this new equation for one of the variables. Let's solve for x:
-9x - 40y = 35
-9x = 35 + 40y
x = (35 + 40y) / -9
Step 6: Substitute the expression for x into one of the original equations. Let's use Equation 1 for this step:
9x + 5y = 35
9((35 + 40y) / -9) + 5y = 35
Simplifying the equation:
(35 + 40y) + 5y = 35
35 + 45y = 35
45y = 0
Step 7: Solve for y:
45y = 0
y = 0 / 45
y = 0
Step 8: Substitute the value of y back into the expression for x:
x = (35 + 40y) / -9
x = (35 + 40(0)) / -9
x = 35 / -9
Step 9: Simplify the expression for x:
x = -35/9
Therefore, the solution to the system of equations is x = -35/9 and y = 0.
To solve the system of equations using the linear combination method, follow these steps:
Step 1: Choose one equation and simplify it by multiplying both sides by a number to make the coefficient of either x or y the same in both equations. In this case, we can simplify the second equation by multiplying it by 9 to make the coefficient of y the same as the first equation:
9(2x+5y) = 9(0)
18x + 45y = 0
Step 2: Now, write the two simplified equations together:
9x + 5y = 35
18x + 45y = 0
Step 3: Multiply the first equation by -2:
-2(9x + 5y) = -2(35)
-18x - 10y = -70
Step 4: Now, write the third equation with the second equation:
-18x - 10y = -70
18x + 45y = 0
Step 5: Add the equations together by aligning like terms:
(-18x + 18x) + (-10y + 45y) = (-70 + 0)
35y = -70
Step 6: Solve for y by dividing both sides of the equation by 35:
y = -70/35
y = -2
Step 7: Substitute the value of y into one of the original equations. Using the first equation:
9x + 5(-2) = 35
9x -10 = 35
Step 8: Solve for x:
9x = 35 + 10
9x = 45
x = 45/9
x = 5
Step 9: Check the solution by substituting the values of x and y into the other original equation. Using the second equation:
2(5) + 5(-2) = 0
10 -10 = 0
0 = 0
Since both sides are equal, the solution (x = 5, y = -2) satisfies the system of equations. Therefore, the solution to the system of equations is x = 5, y = -2.