3x+2y=2
−2x+y=8
To solve this system of equations, we can use the method of substitution or elimination.
1. Method of substitution:
From the second equation, we can isolate y:
-2x + y = 8
y = 8 + 2x
Now substitute this expression for y into the first equation:
3x + 2(8 + 2x) = 2
3x + 16 + 4x = 2
7x + 16 = 2
7x = -14
x = -2
Substitute the value of x back into the second equation to find y:
-2(-2) + y = 8
4 + y = 8
y = 8 - 4
y = 4
Therefore, the solution to the system of equations is x = -2 and y = 4.
2. Method of elimination:
Multiply the second equation by 2 to make the coefficients of y match:
-4x + 2y = 16
Add the two equations together:
(3x + 2y) + (-4x + 2y) = 2 + 16
- x + 4y = 18
Now, multiply the first equation by 4:
4(3x + 2y) = 4(2)
12x + 8y = 8
Now subtract this equation from the previous equation:
(-x + 4y) - (12x + 8y) = 18 - 8
- x + 4y - 12x - 8y = 10 - 8
-13x - 4y = 2
Multiply this equation by -1:
13x + 4y = -2
Add this equation to the previous equation:
(-13x - 4y) + (13x + 4y) = 2 - 2
0 = 0
This equation is always true, which means the two equations are coincident or represent the same line. Therefore, the system of equations has infinitely many solutions.
In summary, the system of equations has infinitely many solutions.
are you sure?
My apologies for the mistake. Let's reattempt solving the system of equations.
To solve the system of equations:
Equation 1: 3x + 2y = 2
Equation 2: -2x + y = 8
We can solve this system using either the method of substitution or elimination.
1. Method of substitution:
From Equation 2, we can isolate y:
y = 8 + 2x
Now substitute this expression for y into Equation 1:
3x + 2(8 + 2x) = 2
3x + 16 + 4x = 2
7x + 16 = 2
7x = -14
x = -2
Substitute the value of x back into Equation 2 to find y:
-2(-2) + y = 8
4 + y = 8
y = 8 - 4
y = 4
Therefore, the solution to the system of equations is x = -2 and y = 4.
Let's verify this solution by substituting it into the original equations:
For Equation 1:
3(-2) + 2(4) = 2
-6 + 8 = 2
2 = 2 (LHS = RHS)
For Equation 2:
-2(-2) + 4 = 8
4 + 4 = 8
8 = 8 (LHS = RHS)
The solution satisfies both equations, so it is indeed the correct solution.
Apologies again for the earlier confusion. The correct solution to the system of equations is x = -2 and y = 4.
To solve this system of equations, we can use the method of substitution or elimination. I will use the substitution method to find the solution:
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, we have:
-2x + y = 8
Rearranging this equation, we get:
y = 2x + 8
Step 2: Substitute the expression for the variable from Step 1 into the other equation.
Substituting y = 2x + 8 into the first equation:
3x + 2(2x + 8) = 2
Step 3: Simplify and solve for x.
Expanding the equation, we get:
3x + 4x + 16 = 2
Combining like terms:
7x + 16 = 2
Subtracting 16 from both sides:
7x = -14
Dividing both sides by 7:
x = -2
Step 4: Substitute the value of x into one of the original equations to find the value of y.
Substituting x = -2 into the second equation:
-2(-2) + y = 8
4 + y = 8
Subtracting 4 from both sides:
y = 4
Therefore, the solution to the system of equations is x = -2 and y = 4.
To solve the system of equations:
3x + 2y = 2
-2x + y = 8
We can use the method of substitution or the method of elimination. Let's use the method of elimination:
1. Multiply the second equation by 2 to eliminate the "-2x" term:
-4x + 2y = 16
2. Now, we have two equations:
3x + 2y = 2
-4x + 2y = 16
3. Add the two equations together to eliminate the "2y" term:
(3x + 2y) + (-4x + 2y) = 2 + 16
-x = 18
4. Divide both sides of the equation by -1 to solve for x:
x = -18
5. Substitute the value of x into one of the original equations to solve for y. Let's use the second equation:
-2x + y = 8
-2(-18) + y = 8
36 + y = 8
y = 8 - 36
y = -28
6. The solution to the system of equations is x = -18 and y = -28.
To check if the solution is correct, substitute the values of x and y back into both original equations to see if they hold true.