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Question
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Solve the system by substitution.
{−4.5x−2y=−12.53.25x−y=−0.75
(3 points)
To solve the system by substitution, we will solve one of the equations for one variable and substitute it into the other equation.
Let's solve the second equation for y:
3.25x - y = -0.75
Add y to both sides of the equation:
3.25x = y - 0.75
Now, let's substitute this expression for y in the first equation:
-4.5x - 2(y - 0.75) = -12.5
Distribute the -2 to both terms inside the parentheses:
-4.5x - 2y + 1.5 = -12.5
Combine like terms:
-4.5x - 2y = -14
Now we have a new equation:
-4.5x - 2y = -14
Let's solve this equation for x:
-4.5x = -2y - 14
Divide both sides of the equation by -4.5:
x = (-2y - 14) / -4.5
Now we can substitute this expression for x in the equation -4.5x - 2y = -14:
-4.5((-2y - 14) / -4.5) - 2y = -14
Simplify:
2y + 14 - 2y = -14
The y terms cancel out:
14 = -14
This equation is not true, which means there is no solution to the system of equations by substitution.
To solve the system of equations by substitution, we can solve one of the equations for one variable and substitute that expression into the other equation. Let's solve the second equation for y and substitute it into the first equation.
Given equations:
-4.5x - 2y = -12.5
3.25x - y = -0.75
Step 1: Solve the second equation for y:
Rearrange the second equation:
y = 3.25x + 0.75
Step 2: Substitute y in the first equation with the expression we found in Step 1:
-4.5x - 2(3.25x + 0.75) = -12.5
Step 3: Simplify the equation:
-4.5x - 6.5x - 1.5 = -12.5
-11.0x - 1.5 = -12.5
Step 4: Move the constant term to the right side:
-11.0x = -12.5 + 1.5
-11.0x = -11.0
Step 5: Divide both sides of the equation by -11.0 to solve for x:
x = -11.0 / -11.0
x = 1.0
Step 6: Substitute the value of x into the expression for y we found in Step 1:
y = 3.25(1.0) + 0.75
y = 3.25 + 0.75
y = 4.0
Therefore, the solution for the system of equations is x = 1.0 and y = 4.0.
To solve the system of equations by substitution, we need to solve one equation for one variable and then substitute that expression into the other equation. Let's start with the second equation.
The second equation is: 3.25x - y = -0.75.
Step 1: Solve the second equation for y:
y = 3.25x + 0.75.
Now we have the expression for y in terms of x. We can substitute this expression into the first equation.
Step 2: Substitute y = 3.25x + 0.75 into the first equation:
-4.5x - 2(3.25x + 0.75) = -12.5.
Step 3: Simplify the equation:
-4.5x - 6.5x - 1.5 = -12.5.
-11.0x - 1.5 = -12.5.
Step 4: Add 1.5 to both sides of the equation:
-11.0x = -11.0.
Step 5: Divide both sides of the equation by -11.0:
x = 1.0.
Now that we have found the value of x, we can substitute it back into the expression for y from step 1.
Step 6: Substitute x = 1.0 into y = 3.25x + 0.75:
y = 3.25(1.0) + 0.75.
y = 3.25 + 0.75.
y = 4.0.
So, the solution to the system of equations is x = 1.0 and y = 4.0.