continued
4c=3d+3 c=d-1 i got (6,7)
x=1/2y+3 2x-y=6 i got no solution
8x+2y=13 4x+y=11 i got no solution
2x-y=-4 -3x+y=-9 i got (13,30)
3x-5y=11 x-3y=1 i got (7,2)
all correct.
To solve each of these systems of equations, you used a method called substitution or elimination. Let's go through each problem and explain how you got the answers.
1. 4c = 3d + 3 and c = d - 1
To solve this system of equations using substitution, you substituted the expression for c from the second equation into the first equation:
4(d - 1) = 3d + 3
Simplifying: 4d - 4 = 3d + 3
Combining like terms: d = 7
Then, to find the value of c, you substituted d = 7 into the second equation:
c = 7 - 1 = 6
So, the solution to this system of equations is (6, 7).
2. x = 1/2y + 3 and 2x - y = 6
This system of equations is a linear system with two variables. To solve it, you need to eliminate one of the variables. In this case, we can eliminate x.
Multiply the first equation by 2 to make the coefficients of x in both equations the same:
2x = y + 6
Now we have:
2x - y = 6
2x = y + 6
Subtracting the second equation from the first equation, we have:
0 = 0
This equation has no variables and is always true. Therefore, the system has no solution.
3. 8x + 2y = 13 and 4x + y = 11
Similar to the previous problem, this is a linear system of equations. To eliminate one variable, we'll multiply the second equation by 2 to make the coefficients of y the same in both equations:
4x + y = 11
8x + 2y = 13
Now we can subtract the first equation from the second equation:
8x + 2y - (4x + y) = 13 - 11
Simplifying: 4x + y = 2
But this is the same as our previous equation for the second equation, so the system of equations is inconsistent and has no solution.
4. 2x - y = -4 and -3x + y = -9
This system can be solved by the method of substitution. Solve one equation for one variable and substitute it into the other equation:
From the first equation, we have: y = 2x + 4
Substituting this expression for y into the second equation:
-3x + (2x + 4) = -9
Simplifying: -3x + 2x + 4 = -9
Combining like terms: -x + 4 = -9
Moving the constants to the other side of the equation: -x = -13
Multiply both sides by -1 to solve for x: x = 13
To find the value of y, substitute x = 13 into the first equation:
2(13) - y = -4
Simplifying: 26 - y = -4
Move the constant to the other side of the equation: -y = -30
Multiply both sides by -1 to solve for y: y = 30
The solution to this system of equations is (13, 30).
5. 3x - 5y = 11 and x - 3y = 1
This system can be solved using either substitution or elimination. Let's use the substitution method.
From the second equation, we have: x = 1 + 3y
Substituting this expression for x into the first equation:
3(1 + 3y) - 5y = 11
Distributing: 3 + 9y - 5y = 11
Combining like terms: 4y + 3 = 11
Moving the constant to the other side of the equation: 4y = 8
Solving for y: y = 2
To find the value of x, substitute y = 2 into the second equation:
x - 3(2) = 1
Simplifying: x - 6 = 1
Move the constant to the other side of the equation: x = 7
Therefore, the solution to this system of equations is (7, 2).
Keep in mind that double-checking your work and verifying your solution by substituting into the original equations can help ensure accuracy in solving systems of equations.