USE LINEAR COMBINATIONS TO SOLVE THE SYSTEM OF LINEAR EQUATIONS (please show steps)
1) -X+8Y=16 2) 4X+5Y=-2 3)4Y=15-3X
3X+4Y=36 5X-4Y=-23 2Y=3X+21
I will do one. Then, I will happily critique your thinking on others. Take the third.
4y=15-3x
2y=3x+21
double the second one..
4y=15-3x
4y=6x+42
subtract, then the result is ..
0=9x-45 and solve for x.
To solve the system of linear equations using linear combinations, follow these steps:
Step 1: Rearrange the equations
- X + 8Y = 16 => X = 8Y + 16 (Equation 1)
4X + 5Y = -2 => X = (-5Y - 2)/4 (Equation 2)
4Y = 15 - 3X => 3X + 4Y = 15 (Equation 3)
2Y = 3X + 21 => -3X + 2Y = 21 (Equation 4)
Step 2: Choose two pairs of equations to eliminate one variable
Let's choose Equation 1 and Equation 2 to eliminate X.
Multiply Equation 1 by 4 to match the denominator in Equation 2:
4(X) = 4(8Y + 16) => 4X = 32Y + 64 (Equation 5)
Now subtract Equation 5 from Equation 2:
(4X + 5Y) - (4X) = (-2) - (32Y + 64)
5Y = -2 - 32Y - 64
5Y + 32Y = -66
37Y = -66
Y = -66 / 37
Y = -1.783 (approx)
Step 3: Substitute the value of Y into one of the original equations to find X
Let's substitute Y = -1.783 into Equation 1:
X = 8(-1.783) + 16
X = -14.264 + 16
X = 1.736 (approx)
Step 4: Check the solutions
Substitute the values of X and Y into the remaining equations to verify they satisfy the system.
Checking Equation 3:
3X + 4Y = 15
3(1.736) + 4(-1.783) = 15
5.208 - 7.132 = 15
15 ≈ 15 (True)
Checking Equation 4:
-3X + 2Y = 21
-3(1.736) + 2(-1.783) = 21
-5.208 - 3.566 = 21
21 ≈ 21 (True)
Therefore, the solution to the system of linear equations is X ≈ 1.736 and Y ≈ -1.783.