Solve by the linear combination method (with or without multiplication).
3x-5y = 7
2x + 3y = 30
To solve the system of equations using the linear combination method, we want to eliminate one variable by adding or subtracting the equations.
Step 1: Let's first multiply each term in the second equation by -3 to get the y-coefficient to be -5y, the same as the first equation.
Multiply the second equation by -3:
-3(2x + 3y) = -3(30)
-6x - 9y = -90
Now we have these two equations:
3x - 5y = 7
-6x - 9y = -90
Step 2: We want to eliminate one variable by adding or subtracting the two equations. In this case, let's eliminate the variable x by multiplying the first equation by 2 and the second equation by 3.
Multiply the first equation by 2:
2(3x - 5y) = 2(7)
6x - 10y = 14
Multiply the second equation by 3:
3(-6x - 9y) = 3(-90)
-18x - 27y = -270
Now we have these two equations:
6x - 10y = 14
-18x - 27y = -270
Step 3: Add the two new equations together to eliminate the variable x.
(6x - 10y) + (-18x - 27y) = 14 + (-270)
-12y = -256
Step 4: Solve for y by dividing both sides of the equation by -12.
-12y / -12 = -256 / -12
y = 21.33 (rounded to two decimal places)
Step 5: Substitute the value of y back into one of the original equations to solve for x. Let's use the first equation:
3x - 5(21.33) = 7
3x - 106.65 = 7
3x = 113.65
x = 37.88 (rounded to two decimal places)
So the solution to the system of equations is x = 37.88 and y = 21.33.
Your last three posts are basically the same type of questions.
I will do this one for you, then you follow that procedure to do the others, ok?
The object is to have the coefficients of either the x or the y to be the same or opposite, so that you can either subtract or add the two equations.
This is based on the concept that by multiplying or dividing each side of an equation by the same number, the equation does not change.
I will go for the x's
#1 times 2: ---> 6x - 10y = 14
#2 times 3: ---> 6x + 9y = 90
subtract them
-19y = -76
divide by -19
y = 4
sub that back into either of the two original equations ....
y = 4 into #2
2x + 3(4) = 30
2x = 30-12
2x = 18
x = 9
check by using those values in the equation that was not used in the second part.
#1 :
Left Side = 3x - 5y
= 3(9) - 5(4)
= 27-20
= 7
= Right Side
So my solution is correct.
x = 9, y = 4