How would you solve these systems? I'm in need help!! Thanks so much!
1. -7x-6y=-21
-x-6y=-3
2. -7x+2y=13
-14x+4y=26
3. 2x+10y=30
-10x-6y=-18
For -7x-6y=-21
Simplifying
-7x + -6y = -21
Solving
-7x + -6y = -21
Solving for variable 'x'.
Move all terms containing x to the left, all other terms to the right.
Add '6y' to each side of the equation.
-7x + -6y + 6y = -21 + 6y
Combine like terms: -6y + 6y = 0
-7x + 0 = -21 + 6y
-7x = -21 + 6y
Divide each side by '-7'.
x = 3 + -0.8571428571y
Simplifying
x = 3 + -0.8571428571y
You would use the same technique for the other ones, also.
for all 3 , elimination looks like the easiest way
1.
-7x-6y=-21
-x-6y=-3
subtract them to eliminate the y's
-6x = -18
x = 3
back into the 2nd:
-3-6y=-3
-6y=0
y = 0 ----------> x = 3, y = 0
2.
-7x+2y=13
-14x+4y=26
Did you notice that the 2nd equation is simply twice the first ?
So there is an infinite number of solutions, no unique solution
3.
2x+10y=30
-10x-6y=-18
first one times 5 ---> 10x + 20y = 60
write down the 2nd --> -10x - 6y = -18
add them:
14y = 42
y = 3
back into the first:
2x + 30 = 30
2x = 0
x=0
To solve these systems of equations, you can use the method of substitution or the method of elimination. I will explain how to solve each system using both methods.
System 1:
-7x - 6y = -21
-x - 6y = -3
Method of Substitution:
Step 1: Solve one equation for one variable in terms of the other variable. Let's solve the second equation for x:
-x = 6y - 3
x = -6y + 3
Step 2: Substitute the expression for x in terms of y into the other equation:
-7(-6y + 3) - 6y = -21
Step 3: Simplify and solve for y:
42y - 21 - 6y = -21
36y = 0
y = 0
Step 4: Substitute the value of y back into one of the original equations to solve for x:
-x - 6(0) = -3
-x = -3
x = 3
So the solution to this system is x = 3 and y = 0.
Method of Elimination:
Step 1: Multiply one or both equations by constants so that the coefficients of one of the variables in both equations are the same, but with opposite signs. This will allow us to eliminate that variable when we add or subtract the equations. In this case, we can multiply the second equation by -7 to get:
7x + 42y = 21
Step 2: Now we can add the two equations together to eliminate y:
-7x - 6y + 7x + 42y = -21 + 21
36y = 0
y = 0
Step 3: Substitute the value of y back into one of the original equations to solve for x:
-x - 6(0) = -3
-x = -3
x = 3
So the solution to this system is x = 3 and y = 0.
Now let's move on to the next systems:
System 2:
-7x + 2y = 13
-14x + 4y = 26
Method of Substitution:
Step 1: Solve one equation for one variable in terms of the other variable. Let's solve the first equation for x:
-7x = -2y + 13
x = (2y - 13)/7
Step 2: Substitute the expression for x in terms of y into the other equation:
-14((2y - 13)/7) + 4y = 26
Step 3: Simplify and solve for y:
-4y + 26 - 4y = 26
-8y = 0
y = 0
Step 4: Substitute the value of y back into one of the original equations to solve for x:
-7x + 2(0) = 13
-7x = 13
x = -13/7
So the solution to this system is x = -13/7 and y = 0.
Method of Elimination:
Step 1: Multiply one or both equations by constants so that the coefficients of one of the variables in both equations are the same, but with opposite signs. In this case, we don't need to make any adjustments as the coefficients of y are already opposites.
Step 2: Add the two equations together to eliminate y:
(-7x + 2y) + (-14x + 4y) = 13 + 26
-21x + 6y = 39
Step 3: Solve for y in terms of x:
6y = 21x - 39
y = (7x - 13)/2
Now we have an equation for y in terms of x.
So the solution to this system would be an infinite number of solutions, where x can take any value and y can be calculated using the equation y = (7x - 13)/2.
Lastly, System 3:
2x + 10y = 30
-10x - 6y = -18
You can try solving this system using both the method of substitution and the method of elimination following the steps mentioned above.