Solve the system by substitution
-4.5x - 2y = -12.5
3.25x - y = -0.75
Please show steps I’m not sure what to do.
Thanks in advance!
To solve the system of equations by substitution method, we will solve one equation for one variable and substitute that expression into the other equation.
Let's solve the second equation for y:
3.25x - y = -0.75
Add y to both sides to isolate the variable:
3.25x - y + y = -0.75 + y
Simplify:
3.25x = y - 0.75
Now, we can substitute this expression for y in the first equation:
-4.5x - 2y = -12.5
Replace y with y - 0.75:
-4.5x - 2(y - 0.75) = -12.5
Distribute the -2:
-4.5x - 2y + 1.5 = -12.5
Combine like terms:
-4.5x - 2y = -12.5 - 1.5
Simplify:
-4.5x - 2y = -14
Now we have the system of equations:
-4.5x - 2y = -14
3.25x - y = -0.75
From here, we can solve this system using any method (substitution, elimination, or matrices). Let’s continue using the substitution method.
To solve for x, let's isolate x in the second equation:
3.25x = y - 0.75
Divide both sides by 3.25 to solve for x:
x = (y - 0.75) / 3.25
Now we can substitute this expression for x in the first equation:
-4.5x - 2y = -14
Replace x with (y - 0.75) / 3.25:
-4.5((y - 0.75) / 3.25) - 2y = -14
Multiply both the numerator and denominator by the common factor 1 / 0.25 to clear the fraction:
-4.5 * (y - 0.75) * (1 / 3.25) - 2y = -14
Simplify:
-1.38461538(y - 0.75) - 2y = -14
Distribute the -1.38461538:
-1.38461538y + 1.03846154 - 2y = -14
Combine like terms:
-1.38461538y - 2y + 1.03846154 = -14
-3.38461538y + 1.03846154 = -14
Subtract 1.03846154 from both sides:
-3.38461538y = -14 - 1.03846154
Simplify:
-3.38461538y = -15.03846154
Finally, divide both sides by -3.38461538 to solve for y:
y = (-15.03846154) / (-3.38461538)
y ≈ 4.4408921
Now that we have the value of y, we can substitute it back into one of the original equations to find the value of x. Let's use the second equation:
3.25x - y = -0.75
Substitute the value of y:
3.25x - 4.4408921 = -0.75
Add 4.4408921 to both sides:
3.25x = -0.75 + 4.4408921
Simplify:
3.25x = 3.6908921
Finally, divide both sides by 3.25 to solve for x:
x = 3.6908921 / 3.25
x ≈ 1.1342921
Therefore, the solution to the system of equations is x ≈ 1.1342921 and y ≈ 4.4408921.
To solve the system of equations by substitution, you need to solve one equation for one variable, and then substitute that expression into the other equation. Let's solve the second equation for y first:
3.25x - y = -0.75
Add y to both sides:
y = 3.25x + 0.75
Now that we have an expression for y in terms of x, we can substitute it into the first equation:
-4.5x - 2(3.25x + 0.75) = -12.5
Simplify the equation:
-4.5x - 6.5x - 1.5 = -12.5
Combine like terms:
-11.0x - 1.5 = -12.5
Add 1.5 to both sides:
-11.0x = -11.0
Divide by -11.0:
x = 1.0
Now, substitute the value of x back into the equation we solved for y:
y = 3.25(1.0) + 0.75
Simplify:
y = 3.25 + 0.75
y = 4.0
Therefore, the solution to the system of equations is x = 1.0 and y = 4.0.
eq #2 says that y = 3.25x + 0.75
So substitute that into eq #1, and you have
-4.5x - 2(3.25x + 0.75) = -12.5
Just solve that for x, and then use that value to get y.