solve the system of equations
7x+3y=-25
-7x+9y=-47
To solve the system of equations, we can use the method of substitution or elimination. I will demonstrate solving it using the substitution method:
1) Solve one equation for one variable in terms of the other variable.
From the first equation, we can solve for x:
7x + 3y = -25
Move 3y to the other side:
7x = -25 - 3y
Divide by 7:
x = (-25 - 3y)/7
2) Substitute the value of x in the second equation.
-7((-25 - 3y)/7) + 9y = -47
Simplify:
-(-25 - 3y) + 9y = -47
25 + 3y + 9y = -47
Combine like terms:
25 + 12y = -47
3) Solve for y:
12y = -47 - 25
12y = -72
Divide by 12:
y = -6
4) Substitute the value of y back into either of the original equations to solve for x. Let's use the first equation:
7x + 3(-6) = -25
7x - 18 = -25
Add 18 to both sides:
7x = -25 + 18
7x = -7
Divide by 7:
x = -1
Thus, the solution to the system of equations is x = -1 and y = -6.
To solve the system of equations:
Step 1: Multiply the first equation by -1 to eliminate x.
-1(7x + 3y) = -1(-25)
-7x - 3y = 25
Step 2: Add this new equation to the second equation to eliminate x.
(-7x + 9y) + (-7x - 3y) = -47 + 25
-14x + 6y = -22
Step 3: Divide the entire equation by -2 to simplify it.
(-14x + 6y)/-2 = -22/-2
7x - 3y = 11
Step 4: Now we have a new system:
7x - 3y = 11
-7x + 9y = -47
Step 5: Add the two equations together to eliminate x.
(7x - 3y) + (-7x + 9y) = 11 + (-47)
6y = -36
Step 6: Solve for y by dividing both sides of the equation by 6.
6y/6 = -36/6
y = -6
Step 7: Substitute the value of y into one of the original equations, let's use the first equation.
7x + 3(-6) = -25
Simplify:
7x - 18 = -25
Step 8: Add 18 to both sides of the equation.
7x - 18 + 18 = -25 + 18
7x = -7
Step 9: Solve for x by dividing both sides of the equation by 7.
7x/7 = -7/7
x = -1
Step 10: The solution to the system of equations is x = -1 and y = -6.
To solve the system of equations:
Step 1: Choose one of the equations and multiply it by a number so that when you add or subtract it with the other equation, one variable will cancel out.
In this case, we can multiply the first equation by 7 to get:
(7)(7x+3y) = (7)(-25)
49x + 21y = -175
Step 2: Add or subtract the two equations to eliminate one variable.
Now, subtract the revamped first equation from the second equation:
-7x + 9y - (49x + 21y) = -47 - (-175)
This simplifies to:
-7x + 9y - 49x - 21y = -47 + 175
Combine like terms:
-56x - 12y = 128
Step 3: Solve the resulting equation for one variable.
Next, we need to isolate one variable (either x or y) in one of the equations. Let's isolate x in the equation we obtained in step 2:
-56x - 12y = 128
First, let's solve this equation for x:
-56x = 128 + 12y
Divide both sides by -56 to find the value of x:
x = (128 + 12y) / -56
Step 4: Substitute the value of x into one of the original equations to solve for the other variable.
Let's substitute the value of x in the first original equation:
7x + 3y = -25
7((128 + 12y) / -56) + 3y = -25
Simplify:
(7 * 128 + 84y) / -56 + 3y = -25
Multiply both sides by -56 to eliminate the fractions:
7 * 128 + 84y + (-56 * 3y) = -25 * -56
Simplify further:
896 + 84y - 168y = 1400
Combine like terms:
-84y = 504
Divide both sides by -84 to solve for y:
y = 504 / -84
y = -6
Step 5: Substitute the value of y back into one of the original equations to solve for x.
Let's use the first original equation:
7x + 3(-6) = -25
7x - 18 = -25
Add 18 to both sides:
7x = -25 + 18
7x = -7
Divide both sides by 7 to solve for x:
x = -7 / 7
x = -1
Therefore, the solution to the system of equations is x = -1 and y = -6.