Solve this system by substitution: 2x-3y=12
x-4y=6
Slve this system by elimination: 8x+5y=22
6x+ 2y=13
In the second equation,
x=4y+6 Put that in the first equation for x, and solve for y.
In the second system, multipy the first equation by 2, and the second by 5
16x+10y=24
30x+10y=44
Now subtract one equation from the other.
Solve this system using substitution. Write your answer as an ordered pair. x = -3 3x – 4y = -37
To solve the system of equations by substitution, follow these steps:
1. Solve one equation for one variable in terms of the other variable.
Let's solve the second equation, x - 4y = 6, for x:
x = 6 + 4y
2. Substitute the expression obtained in step 1 into the other equation.
We substitute x with 6 + 4y in the first equation:
2(6 + 4y) - 3y = 12
3. Simplify and solve the resulting equation for y.
Distribute 2 into the parentheses:
12 + 8y - 3y = 12
Combine like terms:
5y = 0
Divide both sides by 5:
y = 0
4. Substitute the value of y back into one of the original equations to solve for x.
Let's substitute y = 0 into the second equation, x - 4y = 6:
x - 4(0) = 6
x = 6
Therefore, the solution to the system of equations is x = 6 and y = 0.
To solve the system of equations by elimination, follow these steps:
1. Multiply one or both equations (if necessary) by a constant to create a common coefficient for one of the variables.
Let's multiply the first equation, 8x + 5y = 22, by 2 to make the coefficients of y equal in absolute value but with different signs:
16x + 10y = 44
2. Add or subtract the resulting equations to eliminate one variable.
Subtract the second equation from the modified first equation:
(16x + 10y) - (6x + 2y) = 44 - 13
3. Simplify the resulting equation and solve for the remaining variable.
Distribute and combine like terms:
16x + 10y - 6x - 2y = 31
10x + 8y = 31
4. Solve the equation obtained in step 3 for the remaining variable.
Divide by 2:
5x + 4y = 15.5
Now we have a new equation involving x and y, which we can solve using any method of your choice (substitution, elimination, etc.). Let's use the substitution method:
Solve the second equation, 5x + 4y = 15.5, for x:
x = (15.5 - 4y) / 5
5. Substitute the expression obtained in step 4 back into one of the original equations to solve for the other variable.
We substitute x with (15.5 - 4y) / 5 in the first equation, 8x + 5y = 22:
8((15.5 - 4y) / 5) + 5y = 22
Simplify and solve the resulting equation for y:
(62 -16y) / 5 + 5y = 22
Multiply every term by 5 to clear the fraction:
62 - 16y + 25y = 110
Combine like terms:
9y = 48
Divide both sides by 9:
y = 48/9
y = 5.33 (approx)
6. Substitute the value of y back into one of the original equations to solve for x.
Let's substitute y ≈ 5.33 into the first equation, 8x + 5y = 22:
8x + 5(5.33) = 22
Solve for x:
8x + 26.65 = 22
Subtract 26.65 from both sides:
8x = -4.65
Divide both sides by 8:
x = -0.58125 (approx)
Therefore, the solution to the system of equations is x ≈ -0.58125 and y ≈ 5.33.