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Solve the system substitution.
{(-4.5x-2y=-12.5),(3.25x-y=-0.75)
To solve this system of equations by substitution, we will solve one equation for one variable and substitute it into the other equation.
Let's start by solving the second equation for y:
3.25x - y = -0.75
To isolate y, we can subtract 3.25x from both sides:
-y = -0.75 - 3.25x
To make the coefficient of y positive, we can multiply both sides by -1:
y = 0.75 + 3.25x (Equation 1)
Now, we will substitute this expression for y into the first equation:
-4.5x - 2(0.75 + 3.25x) = -12.5
To simplify, we distribute the 2:
-4.5x - 1.5 - 6.5x = -12.5
Combine like terms:
-11x - 1.5 = -12.5
To isolate x, we can add 1.5 to both sides:
-11x = -12.5 + 1.5
-11x = -11
Divide both sides by -11 to solve for x:
x = -11 / -11
x = 1
Now that we have the value of x, we can substitute it back into Equation 1 to find y:
y = 0.75 + 3.25(1)
y = 0.75 + 3.25
y = 4
Therefore, the solution to the system of equations is x = 1 and y = 4.
To solve the system of equations using substitution, we will solve one equation for one variable, then substitute that value into the other equation.
Let's start with equation 1:
-4.5x - 2y = -12.5
We can solve this equation for x.
-4.5x = 2y - 12.5
Divide both sides of the equation by -4.5:
x = (2y - 12.5) / -4.5
Now, we will substitute this value of x into equation 2:
3.25x - y = -0.75
Substitute the value of x in terms of y:
3.25((2y - 12.5) / -4.5) - y = -0.75
Multiply 3.25 by (2y - 12.5):
(6.5y - 40.625) / -4.5 - y = -0.75
Now, let's simplify the equation:
(6.5y - 40.625) / -4.5 - y = -0.75
Multiply through by -4.5 to eliminate the denominator:
6.5y - 40.625 - 4.5y = -3.375
Combine like terms:
2y - 40.625 = -3.375
Add 40.625 to both sides:
2y = 37.25
Divide both sides by 2:
y = 37.25 / 2
Simplify:
y = 18.625
Now, substitute the value of y back into equation 1 to find x:
-4.5x - 2(18.625) = -12.5
-4.5x - 37.25 = -12.5
Add 37.25 to both sides:
-4.5x = 24.75
Divide both sides by -4.5:
x = 24.75 / -4.5
Simplify:
x = -5.5
Therefore, the solution to the system of equations is x = -5.5 and y = 18.625.
To solve this system of equations using substitution, follow these steps:
Step 1: Choose one of the equations and solve for one variable in terms of the other variable.
Let's start with the second equation, 3.25x - y = -0.75. Solve for y:
y = 3.25x + 0.75
Step 2: Substitute the expression we found for y into the other equation.
Replace y in the first equation, -4.5x - 2y = -12.5, with 3.25x + 0.75:
-4.5x - 2(3.25x + 0.75) = -12.5
Step 3: Simplify the equation and solve for x.
-4.5x - 6.5x - 1.5 = -12.5
-11.5x - 1.5 = -12.5
-11.5x = -12.5 + 1.5
-11.5x = -11
x = -11 / -11.5
x = 0.9565 (rounded to 4 decimal places)
Step 4: Substitute the value of x back into one of the original equations to solve for y.
Let's use the first equation, -4.5x - 2y = -12.5:
-4.5(0.9565) - 2y = -12.5
-4.30425 - 2y = -12.5
-2y = -12.5 + 4.30425
-2y = -8.19575
y = -8.19575 / -2
y = 4.097875 (rounded to 6 decimal places)
Therefore, the solution to the system of equations is x = 0.9565 and y = 4.097875.