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!
3. x = -6y
y = 2/3x + 5
y = 2/3(-6y) +5
y = -4y + 5
5y = 5
y = 1
Substitute -6y for x in second equation and solve for y. Insert that value into the first equation and solve for x. Check by inserting both values into the second equation.
Use the same process for the first two equations.
To solve the system of equations by substitution, follow these steps for each given problem:
1. 5x - y = 16 and 5x + 2y = 13:
Step 1: Solve one equation for one variable. Let's solve the first equation for x:
5x = y + 16 ⇒ x = (y + 16)/5.
Step 2: Substitute the expression for x from the first equation into the second equation:
5((y+16)/5) + 2y = 13 ⇒ y + 16 + 2y = 13.
Step 3: Simplify the equation and solve for y:
3y + 16 = 13 ⇒ 3y = -3 ⇒ y = -1.
Step 4: Substitute the value of y back into either of the original equations to find x:
5x - (-1) = 16 ⇒ 5x + 1 = 16 ⇒ 5x = 15 ⇒ x = 3.
Therefore, the solution is x = 3 and y = -1.
2. 3x - 3y = 9 and 2x - 4y = -16:
Step 1: Solve one equation for one variable. Let's solve the first equation for x:
3x = 3y + 9 ⇒ x = (3y + 9)/3.
Step 2: Substitute the expression for x from the first equation into the second equation:
2((3y+9)/3) - 4y = -16 ⇒ 2y + 6 - 4y = -16.
Step 3: Simplify the equation and solve for y:
-2y + 6 = -16 ⇒ -2y = -22 ⇒ y = 11.
Step 4: Substitute the value of y back into either of the original equations to find x:
3x - 3(11) = 9 ⇒ 3x - 33 = 9 ⇒ 3x = 42 ⇒ x = 14.
Therefore, the solution is x = 14 and y = 11.
3. x = -6y and y = (2/3)x + 5:
Step 1: Substitute the value of x from the first equation into the second equation:
y = (2/3)(-6y) + 5.
Step 2: Simplify the equation and solve for y:
y = (-12/3)y + 5 ⇒ y = -4y + 5 ⇒ 5y = 5 ⇒ y = 1.
Step 3: Substitute the value of y back into the first equation to find x:
x = -6(1) ⇒ x = -6.
Therefore, the solution is x = -6 and y = 1.
Note: Always verify the solution by substituting the values of x and y back into the original equations to ensure they satisfy both equations.
To solve a system of equations by substitution, follow these steps:
1. Solve one of the equations for one variable in terms of the other variable. Let's call this equation "Equation A".
For example, in the first set of equations:
Equation A: Solve 5x-y=16 for y: y = 5x - 16.
2. Substitute the expression from Equation A into the other equation in the system. Let's call this Equation B.
For example, in the first set of equations:
Equation B: Substitute y = 5x - 16 into 5x + 2y = 13:
5x + 2(5x - 16) = 13.
3. Simplify and solve the resulting equation for the variable. This will give you the value of one variable.
Continuing with the previous example:
5x + 10x - 32 = 13
15x - 32 = 13
15x = 13 + 32
15x = 45
x = 45/15
x = 3.
4. Substitute the value of the solved variable back into one of the original equations to find the value of the other variable.
Using the first equation, 5x - y = 16, substitute x = 3:
5(3) - y = 16
15 - y = 16
-y = 16 - 15
-y = 1
y = -1.
5. Repeat the process for each set of equations.
Let's apply these steps to each of the sets of equations you provided:
1. 5x - y = 16
5x + 2y = 13
Equation A: Solve the first equation for y: y = 5x - 16.
Equation B: Substitute y = 5x - 16 into the second equation: 5x + 2(5x - 16) = 13.
Now, simplify Equation B:
5x + 10x - 32 = 13
15x - 32 = 13
15x = 13 + 32
15x = 45
x = 45/15
x = 3.
Substitute x = 3 into Equation A:
y = 5(3) - 16
y = 15 - 16
y = -1.
So, the solution to the first set of equations is x = 3, y = -1.
2. 3x - 3y = 9
2x - 4y = -16
Equation A: Solve the first equation for x: x = y + 3.
Equation B: Substitute x = y + 3 into the second equation: 2(y + 3) - 4y = -16.
Now, simplify Equation B:
2y + 6 - 4y = -16
-2y + 6 = -16
-2y = -16 - 6
-2y = -22
y = -22/-2
y = 11.
Substitute y = 11 into Equation A:
x = 11 + 3
x = 14.
So, the solution to the second set of equations is x = 14, y = 11.
3. x = -6y
y = (2/3)x + 5
Equation A: Substitute x = -6y into the second equation: y = (2/3)(-6y) + 5.
Now, simplify Equation A:
y = -4y + 5
5y = 5
y = 1.
Substitute y = 1 into Equation A:
x = -6(1)
x = -6.
So, the solution to the third set of equations is x = -6, y = 1.