7x+3y=−25
−7x+9y=−47
To solve the system of equations:
1) Multiply the first equation by 7 and multiply the second equation by -7 to eliminate the x variable:
7(7x+3y) = 7(-25)
-7(-7x+9y) = -7(-47)
This simplifies to:
49x + 21y = -175
49x - 63y = 329
2) Subtract the second equation from the first equation to solve for y:
(49x + 21y) - (49x - 63y) = (-175) - (329)
49x + 21y - 49x + 63y = -175 - 329
84y = -504
y = -504/84
y = -6
3) 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
7x = -25 + 18
7x = -7
x = -7/7
x = -1
Therefore, the solution to the system of equations is x = -1 and y = -6.
To solve this system of equations, we'll use the method of elimination. Let's start by multiplying the first equation by 9 and the second equation by 3 to make the coefficients of x in both equations opposite.
Equation 1: 7x + 3y = -25 (multiply by 9)
Equation 2: -7x + 9y = -47 (multiply by 3)
Now the equations become:
Equation 1: 63x + 27y = -225
Equation 2: -21x + 27y = -141
Next, let's subtract Equation 2 from Equation 1 to eliminate x:
(63x + 27y) - (-21x + 27y) = -225 - (-141)
Simplifying:
63x + 27y + 21x - 27y = -225 + 141
84x = -84
Dividing both sides of the equation by 84:
x = -1
Now that we have the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use Equation 1:
7x + 3y = -25
7(-1) + 3y = -25
-7 + 3y = -25
3y = -25 + 7
3y = -18
Dividing both sides of the equation by 3:
y = -6
So, the solution to the system of equations is x = -1 and y = -6.
To solve this system of equations, you can use the method of substitution or the method of elimination. I will explain both methods, and you can choose which one you prefer.
Method 1: Substitution
1. Solve one equation for one variable in terms of the other variable. Let's solve the first equation for x:
7x + 3y = -25
Subtract 3y from both sides:
7x = -3y - 25
Divide both sides by 7:
x = (-3y - 25)/7
2. Substitute the expression for x obtained in step 1 into the second equation:
-7((-3y - 25)/7) + 9y = -47
Simplify:
3y + 25 + 9y = -47
Combine like terms:
12y + 25 = -47
Subtract 25 from both sides:
12y = -72
Divide both sides by 12:
y = -6
3. Substitute the value of y obtained in step 2 back into the equation from step 1 to find x:
x = (-3*(-6) - 25)/7
Simplify:
x = (18 - 25)/7
x = -7/7
x = -1
Therefore, the solution to the system of equations is x = -1 and y = -6.
Method 2: Elimination
1. Multiply the first equation by 7 and the second equation by -7 to eliminate x:
7(7x + 3y) = 7(-25)
-7(-7x + 9y) = -7(-47)
Simplify:
49x + 21y = -175
49x - 63y = 329
2. Add the resulting equations together to eliminate x:
(49x + 21y) + (49x - 63y) = -175 + 329
Simplify:
98x - 42y = 154
3. Divide both sides of the resulting equation by 14 to simplify:
(98x - 42y)/14 = 154/14
Simplify:
7x - 3y = 11
Now we have a new system of equations:
7x - 3y = 11
-7x + 9y = -47
4. Add the two new equations together to eliminate x again:
(7x - 3y) + (-7x + 9y) = 11 + (-47)
Simplify:
6y = -36
Divide both sides by 6:
y = -6
5. Substitute the value of y back into one of the original equations to find x. Let's use the first equation:
7x + 3(-6) = -25
Simplify:
7x - 18 = -25
Add 18 to both sides:
7x = -7
Divide both sides by 7:
x = -1
Again, we find that the solution to the system of equations is x = -1 and y = -6.
Both methods yield the same solution, so you can use either one.