how do I solve -15x-2y=-31 4x+6)=11 using linear combination method?
To solve the system of equations -15x-2y=-31 and 4x+6y=11 using the linear combination method, follow these steps:
Step 1: Multiply one or both equations by a constant, if necessary, so that the coefficients of one of the variables become opposites.
In this case, let's multiply the second equation by -3 to make the coefficient of y in both equations -6:
-15x - 2y = -31
-3(4x + 6y) = -3(11)
Simplifying the second equation, we get:
-12x - 18y = -33
Now we have two equations with opposite coefficients of y.
Step 2: Add or subtract the two equations to eliminate one variable.
To eliminate y, add the two equations together:
(-15x - 2y) + (-12x - 18y) = (-31) + (-33)
-15x - 2y - 12x - 18y = -64
Simplifying, we get:
-27x - 20y = -64
Step 3: Solve the resulting equation for the remaining variable.
Now, we have a new equation: -27x - 20y = -64
To solve for x, isolate x by adding 20y to both sides of the equation:
-27x - 20y + 20y = -64 + 20y
-27x = -64 + 20y
Finally, divide both sides of the equation by -27 to solve for x:
x = (-64 + 20y) / -27
Step 4: Substitute the found value of x (from step 3) into one of the original equations to solve for y.
Let's substitute the value of x into the first equation, -15x - 2y = -31:
-15((-64 + 20y) / -27) - 2y = -31
Simplify the equation:
(960 - 300y) / 27 - 2y = -31
To simplify further and solve for y, multiply both sides of the equation by 27 to get rid of the denominator:
960 - 300y - 54y = -31 * 27
960 - 354y = -837
Next, isolate y by subtracting 960 from both sides:
-354y = -837 - 960
-354y = -1797
Finally, divide both sides by -354 to solve for y:
y = -1797 / -354
Simplifying:
y ≈ 5.07
Step 5: Substitute the value of y (from step 4) back into one of the original equations to solve for x.
Choosing the first equation, -15x - 2y = -31, substitute the value of y:
-15x - 2(5.07) = -31
Simplify the equation:
-15x - 10.14 = -31
To isolate x, add 10.14 to both sides:
-15x = -31 + 10.14
-15x = -20.86
Finally, divide both sides by -15 to solve for x:
x ≈ 1.39
So, the solution to the system of equations is approximately x = 1.39 and y ≈ 5.07.