Solve system of equations algebraically.
1. 3x-3y=24 and 2x+5y=-37
2. -9x+10y=-2 and 45x-50y=11
3. -9x-3y=-3 and 18x+6y=6
4. -5x-6y=-23 and 8x+7y=29
5. 9x-7y=4 and 5x+3y=-46
I will do the last one, you do the others
5.
9x - 7y = 4
multiply each term by 3 ---> 27x - 21y = 12
5x + 3y = -46
multiply each term by 7 ---> 35x + 21y = -322
add these last two equations
62x = -310
x = -5
sub into
5(-5) + 3y = -46
3y=-21
y = -7
x= -5, y = -7
To solve these systems of equations algebraically, we will use the method of elimination or substitution.
1. 3x - 3y = 24 and 2x + 5y = -37
First, let's eliminate one variable to solve for the other. Multiply the first equation by 2 and the second equation by 3 to make the coefficients of y cancel each other out:
6x - 6y = 48
6x + 15y = -111
Now, subtract the first equation from the second equation to eliminate x:
(6x + 15y) - (6x - 6y) = -111 - 48
21y = -159
y = -159 / 21
y = -7
Substitute this value of y back into one of the original equations:
3x - 3(-7) = 24
3x + 21 = 24
3x = 3
x = 3 / 3
x = 1
The solution to this system is x = 1 and y = -7.
2. -9x + 10y = -2 and 45x - 50y = 11
Let's use the elimination method again to eliminate one variable. Multiply the first equation by 5 and the second equation by 2:
-45x + 50y = -10
90x - 100y = 22
Now, add the two equations together to eliminate y:
(-45x + 50y) + (90x - 100y) = -10 + 22
-45x + 90x = 12
45x = 12
x = 12 / 45
x = 4 / 15
Substitute this value of x back into one of the original equations:
-9(4/15) + 10y = -2
-36/15 + 10y = -2
-12/5 + 10y = -2
10y = -2 + 12/5
10y = -2(5/5) + 12/5
10y = -10/5 + 12/5
10y = 2/5
y = 2/5 / 10
y = 1/25
The solution to this system is x = 4/15 and y = 1/25.
3. -9x - 3y = -3 and 18x + 6y = 6
To eliminate a variable, multiply the first equation by -2 and the second equation by 1:
18x + 6y = 6
18x + 6y = 6
Notice that both equations are equivalent, which means they represent the same line. There are infinite solutions to this system, and any values of x and y that satisfy the equation 18x + 6y = 6 will be a solution.
4. -5x - 6y = -23 and 8x + 7y = 29
Let's multiply the first equation by 8 and the second equation by 5 to eliminate x:
-40x - 48y = -184
40x + 35y = 145
Now, add the two equations together to eliminate x:
(-40x - 48y) + (40x + 35y) = -184 + 145
-13y = -39
y = -39 / -13
y = 3
Substitute this value of y back into one of the original equations:
-5x - 6(3) = -23
-5x - 18 = -23
-5x = -23 + 18
-5x = -5
x = -5 / -5
x = 1
The solution to this system is x = 1 and y = 3.
5. 9x - 7y = 4 and 5x + 3y = -46
To eliminate a variable, multiply the first equation by 3 and the second equation by 7:
27x - 21y = 12
35x + 21y = -322
Now, add the two equations together to eliminate y:
(27x - 21y) + (35x + 21y) = 12 - 322
62x = -310
x = -310 / 62
x = -5
Substitute this value of x back into one of the original equations:
9(-5) - 7y = 4
-45 - 7y = 4
-7y = 4 + 45
-7y = 49
y = 49 / -7
y = -7
The solution to this system is x = -5 and y = -7.