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=−12.5
3.25x−y=−0.75
(3 points)
To solve the system by substitution, we need to solve one equation for one variable and then substitute this expression into the other equation.
First, let's solve the second equation for y:
3.25x - y = -0.75
y = 3.25x + 0.75
Now we can substitute this expression for y into the first equation:
-4.5x - 2(3.25x + 0.75) = -12.5
-4.5x - 6.5x - 1.5 = -12.5
-11.5x - 1.5 = -12.5
Next, let's isolate the x variable:
-11.5x = -12.5 + 1.5
-11.5x = -11
x = -11 / -11.5
x = 0.9565 (rounded to four decimal places)
Now that we have the value of x, we can substitute it back into the expression for y in the second equation:
y = 3.25(0.9565) + 0.75
y = 3.1 + 0.75
y = 3.85
The solution to the system of equations is x = 0.9565 and y = 3.85.
To solve the system of equations by substitution, we will isolate one of the variables in one equation and substitute it into the other equation. Let's start solving the system:
Step 1: Solve the first equation for x or y.
Let's solve the first equation for y:
-4.5x - 2y = -12.5
First, we will move the -2y term to the other side of the equation by adding 2y to both sides:
-4.5x = -12.5 + 2y
Next, we will isolate y by dividing the entire equation by -2:
(1/-2)(-4.5x) = (1/-2)(-12.5 + 2y)
2.25x = 6.25 - y
Now, let's rearrange the equation to get y in terms of x:
2.25x + y = 6.25
Step 2: Substitute the expression for y into the second equation.
The second equation is:
3.25x - y = -0.75
Now we will substitute the expression for y we found in step 1 into the second equation:
3.25x - (2.25x + y) = -0.75
Step 3: Solve the resulting equation for x.
Simplify the equation by distributing the negative sign:
3.25x - 2.25x - y = -0.75
Combine like terms:
1x - y = -0.75
Step 4: Solve the resulting equation for y.
Add y to both sides of the equation:
x - y + y = -0.75 + y
Simplify:
x = -0.75 + y
Step 5: Solve for y in terms of x.
Rearrange the equation to get y in terms of x:
y = x + 0.75
Now we have expressed y in terms of x.
Therefore, the solution to the system of equations is:
x = any real number
y = x + 0.75