solve by substitution. Please show all work.
-6x+5y=1
6x+4y=10
To solve the system of equations by substitution, we can solve one equation for one variable and substitute it into the other equation. Let's begin:
Step 1: Solving the first equation for x.
-6x + 5y = 1
We isolate x by adding 6x to both sides:
5y = 6x + 1
Now, divide both sides of the equation by 5:
y = (6x + 1) / 5
Step 2: Substituting the value of y in the second equation.
6x + 4y = 10
Substitute the value of y from the first equation into the second equation:
6x + 4((6x + 1) / 5) = 10
Step 3: Solve for x.
To simplify the equation, we can multiply both sides by 5 to get rid of the fraction:
5(6x) + 5(4((6x + 1) / 5)) = 5(10)
Simplify further:
30x + 4(6x + 1) = 50
Distribute 4:
30x + 24x + 4 = 50
Combine like terms:
54x + 4 = 50
Now, subtract 4 from both sides:
54x = 46
Finally, divide both sides by 54 to solve for x:
x = 46 / 54
x = 23 / 27
Step 4: Solve for y.
Now that we have the value of x, substitute it back into the equation we found in Step 1:
y = (6x + 1) / 5
y = (6(23/27) + 1) / 5
y = (138/27 + 1) / 5
y = (138/27 + 27/27) / 5
y = (165/27) / 5
y = (165/27) * (1/5)
y = 165/135
y = 11/9
Thus, the solution to the system of equations is x = 23/27 and y = 11/9.
To solve the system of equations -6x + 5y = 1 and 6x + 4y = 10 by substitution, we first need to isolate one variable in either of the equations. Let's solve for x in terms of y in the first equation.
-6x + 5y = 1
Add 6x to both sides:
5y = 6x + 1
Divide both sides by 5:
y = (6/5)x + 1/5
Now that we have an expression for y in terms of x, we can substitute this into the second equation:
6x + 4((6/5)x + 1/5) = 10
Simplify the equation:
6x + (24/5)x + 4/5 = 10
To get rid of the fractions, we can multiply the entire equation by 5:
5(6x) + 5(24/5)x + 5(4/5) = 5(10)
30x + 24x + 4 = 50
Combine like terms:
54x + 4 = 50
Subtract 4 from both sides:
54x = 46
Divide both sides by 54:
x = 46/54
Simplify the fraction:
x = 23/27
Now that we have found the value of x, we can substitute it back into either of the original equations to solve for y. We'll substitute it into the first equation:
-6(23/27) + 5y = 1
Simplify the equation:
-46/27 + 5y = 1
To get rid of the fraction, we'll multiply the entire equation by 27:
27(-46/27) + 27(5y) = 27(1)
-46 + 135y = 27
Add 46 to both sides:
135y = 73
Divide both sides by 135:
y = 73/135
Simplify the fraction:
y = 29/54
So the solutions for the system of equations -6x + 5y = 1 and 6x + 4y = 10 are x = 23/27 and y = 29/54.
from eq #1, 6x = 5y-1
sub that into #2, and you get
5y-1 + 4y = 10
9y = 11
y = 11/9
Now you can get x