Answer:
To solve the system of equations 2y+x=8 and 1+y=2x, we can use the method of substitution or elimination.
Method 1: Substitution
Step 1: Solve one of the equations for one variable in terms of the other variable. Let's solve the second equation for y:
1 + y = 2x
Subtract 1 from both sides:
y = 2x - 1
Step 2: Substitute the expression for y from step 1 into the other equation:
2y + x = 8
Replace y with 2x - 1:
2(2x - 1) + x = 8
Simplify:
4x - 2 + x = 8
Combine like terms:
5x - 2 = 8
Step 3: Solve the resulting equation for x:
5x - 2 = 8
Add 2 to both sides:
5x = 10
Divide both sides by 5:
x = 2
Step 4: Substitute the value of x back into one of the original equations to find y. Let's use the first equation:
2y + x = 8
Replace x with 2:
2y + 2 = 8
Subtract 2 from both sides:
2y = 6
Divide both sides by 2:
y = 3
Therefore, the solution to the system of equations is x = 2 and y = 3.
Method 2: Elimination
Step 1: Multiply one or both of the equations by a constant to make the coefficients of one variable equal in both equations. Let's multiply the second equation by 2 to eliminate the y terms:
2(1 + y) = 2(2x)
2 + 2y = 4x
Step 2: Subtract the resulting equation from the first equation to eliminate the y terms:
(2y + x) - (2 + 2y) = 8 - 4x
2y + x - 2 - 2y = 8 - 4x
Simplify:
x - 2 = 8 - 4x
Step 3: Solve the resulting equation for x:
x - 2 = 8 - 4x
Add 4x to both sides:
5x - 2 = 8
Add 2 to both sides:
5x = 10
Divide both sides by 5:
x = 2
Step 4: Substitute the value of x back into one of the original equations to find y. Let's use the first equation:
2y + x = 8
Replace x with 2:
2y + 2 = 8
Subtract 2 from both sides:
2y = 6
Divide both sides by 2:
y = 3
Therefore, the solution to the system of equations is x = 2 and y = 3
Explanation:
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