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Solve the system substitution.
-4.5x-2y=-12.5
3.25x-y=-0.75
To solve the system of equations by substitution, we can solve one equation for one variable and substitute it into the other equation.
Let's solve the first equation for x:
-4.5x - 2y = -12.5
We can isolate x by moving the -2y term to the other side of the equation:
-4.5x = -12.5 + 2y
Divide both sides of the equation by -4.5:
x = (-12.5 + 2y) / -4.5
Now we substitute this expression for x into the second equation:
3.25x - y = -0.75
Replace x with the expression we found: (-12.5 + 2y) / -4.5
3.25((-12.5 + 2y) / -4.5) - y = -0.75
Now we can simplify this equation and solve for y:
First, distribute 3.25 to (-12.5 + 2y):
((-12.5 * 3.25 + 2y * 3.25) / -4.5) - y = -0.75
Simplify the numerator:
((-40.625+6.5y) / -4.5) - y = -0.75
Now, multiply through by -4.5 to clear the fraction:
-40.625 + 6.5y + -4.5y = -0.75 * -4.5
Simplify the right side:
-40.625 + 2y = 3.375
Move -40.625 to the other side:
2y = 3.375 + 40.625
Simplify the right side:
2y = 44
Divide both sides by 2 to solve for y:
y = 22
Now substitute this value for y into the first equation to find x:
-4.5x - 2(22) = -12.5
Simplify:
-4.5x - 44 = -12.5
Move -44 to the other side:
-4.5x = 44 - 12.5
Simplify the right side:
-4.5x = 31.5
Divide both sides by -4.5 to solve for x:
x = 31.5 / -4.5
Simplify:
x = -7
So the solution to the system of equations is x = -7 and y = 22.
To solve the system of equations using substitution, we will solve one equation for one variable, then substitute this expression into the other equation.
Let's solve the second equation, 3.25x - y = -0.75, for y:
y = 3.25x + 0.75
Now, substitute this expression for y into the first equation, -4.5x - 2y = -12.5:
-4.5x - 2(3.25x + 0.75) = -12.5
Simplify:
-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 this value of x back into y = 3.25x + 0.75:
y = 3.25(1.0) + 0.75
Calculate:
y = 3.25 + 0.75
y = 4.0
Therefore, the solution to the system of equations is x = 1.0 and y = 4.0.
To solve the system of equations using substitution, we need to isolate one variable in one equation and substitute it into the other equation. Let's solve the system of equations step by step.
Step 1: Start with the first equation -4.5x-2y=-12.5.
Step 2: Isolate one variable. In this case, let's isolate y. We can do this by adding 4.5x to both sides of the equation:
-2y = 4.5x - 12.5
Step 3: Divide both sides of the equation by -2 to solve for y:
y = (4.5x - 12.5) / -2
Step 4: Now, let's substitute this expression for y in the second equation 3.25x - y = -0.75:
3.25x - ((4.5x - 12.5) / -2) = -0.75
Step 5: Simplify the equation:
3.25x + (4.5x - 12.5) / 2 = -0.75
Step 6: Multiply both sides of the equation by 2 to get rid of the fraction:
2 * (3.25x + (4.5x - 12.5) / 2) = 2 * -0.75
Step 7: Distribute the 2 to both terms in the parentheses:
6.5x + 4.5x - 12.5 = -1.5
Step 8: Combine like terms:
11x - 12.5 = -1.5
Step 9: Add 12.5 to both sides to isolate x:
11x = 11
Step 10: Divide both sides of the equation by 11 to solve for x:
x = 11 / 11
x = 1
Step 11: Now we have obtained the value of x, we can substitute it back into one of the equations to solve for y. Let's use the first equation:
-4.5(1) - 2y = -12.5
Step 12: Simplify the equation:
-4.5 - 2y = -12.5
Step 13: Add 4.5 to both sides of the equation to isolate y:
-2y = -12.5 + 4.5
Step 14: Simplify the equation:
-2y = -8
Step 15: Divide both sides of the equation by -2 to solve for y:
y = -8 / -2
y = 4
Therefore, the solution to the system of equations is x = 1 and y = 4.