Question
3x-y=4
x+2y=6
solve the system of equations
x+2y=6
solve the system of equations
Answers
Answered by
GPT 3.5
To solve the system of equations, we can use either the substitution method or the elimination method.
Using the substitution method:
1. Solve one of the equations for one variable in terms of the other variable. Let's solve the second equation for x: x = 6 - 2y.
2. Substitute this expression for x in the other equation: 3(6 - 2y) - y = 4.
3. Simplify the equation: 18 - 6y - y = 4.
4. Combine like terms: -7y + 18 = 4.
5. Move the constant term to the other side: -7y = 4 - 18 = -14.
6. Divide both sides by -7 to solve for y: y = -14 / -7 = 2.
7. Substitute this value of y back into one of the original equations: x + 2(2) = 6.
8. Solve for x: x + 4 = 6.
9. Subtract 4 from both sides: x = 6 - 4 = 2.
Therefore, the solution to the system of equations is x = 2 and y = 2.
Using the elimination method:
1. Multiply the first equation by 2 and the second equation by 3 to make the coefficient of y in both equations the same.
The new equations become: 6x - 2y = 8 and 3x + 6y = 18, respectively.
2. Now add the two equations together to eliminate y: (6x - 2y) + (3x + 6y) = 8 + 18.
3. Simplify the equation: 9x + 4y = 26.
4. Divide the equation by 4 to solve for x: 9x/4 + 4y/4 = 26/4.
5. Simplify: 9x/4 + y = 6.5.
From here, we have a value for x in terms of y. Let's solve the second equation for x: x = 6 - 2y.
Now we have two expressions for x. Equating them: x = 9x/4.
Multiply both sides by 4 to eliminate the fraction: 4x = 9x.
Subtract 9x from both sides: 4x - 9x = 0.
Combine like terms: -5x = 0.
Divide by -5: x = 0.
Substituting this value of x into the second equation: 0 + 2y = 6.
Solve for y: 2y = 6.
Divide by 2: y = 3.
Thus, the solution to the system of equations is x = 0 and y = 3.
Using the substitution method:
1. Solve one of the equations for one variable in terms of the other variable. Let's solve the second equation for x: x = 6 - 2y.
2. Substitute this expression for x in the other equation: 3(6 - 2y) - y = 4.
3. Simplify the equation: 18 - 6y - y = 4.
4. Combine like terms: -7y + 18 = 4.
5. Move the constant term to the other side: -7y = 4 - 18 = -14.
6. Divide both sides by -7 to solve for y: y = -14 / -7 = 2.
7. Substitute this value of y back into one of the original equations: x + 2(2) = 6.
8. Solve for x: x + 4 = 6.
9. Subtract 4 from both sides: x = 6 - 4 = 2.
Therefore, the solution to the system of equations is x = 2 and y = 2.
Using the elimination method:
1. Multiply the first equation by 2 and the second equation by 3 to make the coefficient of y in both equations the same.
The new equations become: 6x - 2y = 8 and 3x + 6y = 18, respectively.
2. Now add the two equations together to eliminate y: (6x - 2y) + (3x + 6y) = 8 + 18.
3. Simplify the equation: 9x + 4y = 26.
4. Divide the equation by 4 to solve for x: 9x/4 + 4y/4 = 26/4.
5. Simplify: 9x/4 + y = 6.5.
From here, we have a value for x in terms of y. Let's solve the second equation for x: x = 6 - 2y.
Now we have two expressions for x. Equating them: x = 9x/4.
Multiply both sides by 4 to eliminate the fraction: 4x = 9x.
Subtract 9x from both sides: 4x - 9x = 0.
Combine like terms: -5x = 0.
Divide by -5: x = 0.
Substituting this value of x into the second equation: 0 + 2y = 6.
Solve for y: 2y = 6.
Divide by 2: y = 3.
Thus, the solution to the system of equations is x = 0 and y = 3.