how do you solve this by substitution:
step by step
1. 5x-y=16
5x+2y=13
2. 3x-3y=9
2x-4y=-16
3. x=-6y
y=2/3x+5
thank you!
Sure! I can help you solve the equations step-by-step using the substitution method.
1. 5x - y = 16 ----(Equation 1)
5x + 2y = 13 ----(Equation 2)
To solve this system of equations by substitution, we'll start by solving Equation 1 for one variable (either x or y) and then substitute that expression into Equation 2.
From Equation 1, we can isolate y by subtracting 5x from both sides:
y = 5x - 16
Now, substitute this expression for y into Equation 2:
5x + 2(5x - 16) = 13 (replace y with 5x - 16)
Expanding the parentheses:
5x + 10x - 32 = 13
Combine like terms:
15x - 32 = 13
Next, isolate x by adding 32 to both sides:
15x = 45
Divide both sides by 15 to solve for x:
x = 3
Now that we have the value of x, we can substitute it back into Equation 1 or Equation 2 to solve for y.
Using Equation 1:
5(3) - y = 16
Simplifying:
15 - y = 16
To isolate y, subtract 15 from both sides:
-y = 1
Multiply both sides by -1 to solve for y:
y = -1
So, the solution to the system of equations is x = 3 and y = -1.
Now let's move on to the next system of equations.
2. 3x - 3y = 9 ----(Equation 1)
2x - 4y = -16 ----(Equation 2)
To solve this system using the substitution method, we'll again solve Equation 1 for one variable and substitute it into Equation 2.
From Equation 1, we can solve for x by isolating it:
3x = 3y + 9
x = (3y + 9) / 3
x = y + 3
Now, substitute this expression for x into Equation 2:
2(y + 3) - 4y = -16 (replace x with y + 3)
Expanding the parentheses:
2y + 6 - 4y = -16
Combine like terms:
-2y + 6 = -16
Isolate y by subtracting 6 from both sides:
-2y = -22
Divide both sides by -2:
y = 11
Now substitute this value of y back into Equation 1 to solve for x:
3x - 3(11) = 9
Simplifying:
3x - 33 = 9
Add 33 to both sides:
3x = 42
Divide both sides by 3 to solve for x:
x = 14
So, the solution to the second system of equations is x = 14 and y = 11.
Moving on to the last system of equations:
3. x = -6y ----(Equation 1)
y = (2/3)x + 5 ----(Equation 2)
This set of equations is already solved for one variable. We can substitute Equation 1 into Equation 2 for x.
Using Equation 1, we'll substitute -6y for x in Equation 2:
y = (2/3)(-6y) + 5 (replace x with -6y)
Simplifying:
y = -4y + 5
Add 4y to both sides:
5y = 5
Divide both sides by 5 to solve for y:
y = 1
Now, substitute this value of y back into Equation 1 to solve for x:
x = -6(1)
x = -6
So, the solution to the third system of equations is x = -6 and y = 1.
I hope this helps! Let me know if you have any other questions.
To solve a system of equations by substitution, you follow these steps:
1. Start by solving one of the equations for one variable in terms of the other variable.
Let's solve the first set of equations:
Step 1: Solve equation 1 for x:
5x - y = 16
5x = y + 16
x = (y + 16)/5
2. Substitute the expression for the variable from step 1 into the other equation of the system.
Now, substitute the value of x from step 1 into equation 2:
5x + 2y = 13
5((y + 16)/5) + 2y = 13
Simplify:
y + 16 + 2y = 13
3y + 16 = 13
3y = 13 - 16
3y = -3
y = -3/3
y = -1
3. Substitute the value of the solved variable back into one of the original equations and solve for the other variable.
Now, substitute y = -1 into equation 1:
5x - (-1) = 16
5x + 1 = 16
5x = 16 - 1
5x = 15
x = 15/5
x = 3
4. Verify the solution by substituting the values of x and y back into both original equations.
Substitute x = 3 and y = -1 into equation 1:
5(3) - (-1) = 16
15 + 1 = 16
16 = 16 (True)
Substitute x = 3 and y = -1 into equation 2:
5(3) + 2(-1) = 13
15 - 2 = 13
13 = 13 (True)
Therefore, the solution to the first set of equations is x = 3 and y = -1.
Using the same process, you can solve the other sets of equations provided following the same four steps.