I need help learning to solve substitution.
I have a problem that says
solve the system by substitution
-4.5x-2y=-12.5
3.25x-y=-0.75
Look over your equations and see if any contain a single x or y, that makes it easier.
in your case, the second:
3.25x-y=-0.75
-y = -.75 - 3.25x
y = .75 + 3.25x
-4.5x-2y=-12.5
I would clean it up by multiplying each term by -2
9x + 4y = 25
now sub the first part into this equation.
9x + 4(.75 + 3.25x) = 25
9x + 3 + 13x = 25
22x = 22
x = 1
plug that back into y = .75 + 3.25x
y = .75 + 3.25(1) = 4
so x=1 and y = 4
OR
we have the ordered pair (1,4)
substitution means solving for one variable in terms of another , then substituting the solution into a separate equation
in the 2nd equation
... adding y ... 3.25x = y - 0.75
... adding 0.75 ... 3.25x + 0.75 = y
now substitute the value of y into the 1st equation
-4.5x - 2(3.25x + 0.75) = -12.5
distributing ... -4.5x - 6.5x - 1.5 = -12.5
collecting like terms ... -11x - 1.5 = -12.5
adding 1.5 ... -11x = -11
dividing by -11 ... x = 1
substitute the value of x into a previous equation to find the value of y
To solve the given system of equations by substitution, we will solve one equation for one variable and then substitute it into the other equation. Let's start by solving the first equation for 'x'.
Equation 1: -4.5x - 2y = -12.5
Step 1: We'll isolate 'x' by subtracting '2y' from both sides of the equation:
-4.5x = -12.5 + 2y
Step 2: Next, we'll divide through by -4.5 to solve for 'x':
x = (-12.5 + 2y) / -4.5
Now that we have an expression for 'x', we can substitute it into the second equation.
Equation 2: 3.25x - y = -0.75
Step 1: Replace 'x' with the expression we found for 'x':
3.25((-12.5 + 2y) / -4.5) - y = -0.75
Now we can simplify and solve for 'y'.
Step 2: Distribute 3.25 to the terms inside the parentheses:
((-12.5 + 2y) / -4.5) * 3.25 - y = -0.75
Step 3: Simplify the expression:
(-12.5 + 2y) * 3.25 / -4.5 - y = -0.75
Step 4: Multiply (-12.5 + 2y) by 3.25 and divide by -4.5:
(-40.625 + 6.5y) / -4.5 - y = -0.75
Step 5: Multiply each term by -4.5 to eliminate the fraction:
(-40.625 + 6.5y) - 4.5y = -0.75 * -4.5
Step 6: Simplify and solve for 'y':
-40.625 + 6.5y - 4.5y = 3.375
Step 7: Combine like terms:
2y - 40.625 = 3.375
Step 8: Add 40.625 to both sides:
2y = 3.375 + 40.625
Step 9: Simplify:
2y = 44
Step 10: Divide by 2 to solve for 'y':
y = 44 / 2
Therefore, y = 22.
Now that we have the value of 'y', we can substitute it back into the expression we obtained for 'x' in Equation 1 to find 'x'.
x = (-12.5 + 2(22)) / -4.5
Simplifying further:
x = (-12.5 + 44) / -4.5
x = 31.5 / -4.5
x = -7
To solve the system of equations by substitution, we will follow these steps:
Step 1: Choose one equation and solve it for one variable in terms of the other.
Step 2: Substitute the expression found in the previous step into the other equation.
Step 3: Solve the resulting equation for the remaining variable.
Step 4: Substitute the value found in Step 3 back into one of the original equations to solve for the other variable.
Step 5: Check your solution by substituting the values back into both original equations.
Let's solve the given system of equations step by step:
Step 1: Choose one equation and solve it for one variable in terms of the other.
Let's choose the second equation:
3.25x - y = -0.75
Solving this equation for y, we get:
y = 3.25x + 0.75
Step 2: Substitute the expression found in the previous step into the other equation.
Substitute y = 3.25x + 0.75 into the first equation:
-4.5x - 2(3.25x + 0.75) = -12.5
Step 3: Solve the resulting equation for the remaining variable.
Expand and simplify the equation:
-4.5x - 6.5x - 1.5 = -12.5
-11.5x - 1.5 = -12.5
Adding 1.5 to both sides:
-11.5x = -11
Dividing both sides by -11.5:
x = 1
Step 4: Substitute the value found in Step 3 back into one of the original equations to solve for the other variable.
Substitute x = 1 into the second equation:
3.25(1) - y = -0.75
Simplifying, we get:
3.25 - y = -0.75
Subtracting 3.25 from both sides:
-y = -4
Dividing both sides by -1:
y = 4
Step 5: Check your solution by substituting the values back into both original equations:
Substituting x = 1 and y = 4 into the first equation:
-4.5(1) - 2(4) = -12.5
-4.5 - 8 = -12.5
-12.5 = -12.5
The first equation is true.
Substituting x = 1 and y = 4 into the second equation:
3.25(1) - 4 = -0.75
3.25 - 4 = -0.75
-0.75 = -0.75
The second equation is also true.
Therefore, the solution to the system of equations is x = 1 and y = 4.