Solve each of the following systems, if possible. Indicate whether the system has a unique solution, infinitely many solutions, or no solution.
3x+4y=-17
2x+3y=-13
Please help ASAP I'm having a hard time doing this problem. I can either use substitution or elimination in this problem.
Elimination.
6x + 8y = -34
6x + 9y = -39
Subtracting the first from the second, gives you y = -5
Use that in one of the original equations to find x.
You should use elimination.
Firstly, multiply both equations so that the coefficients of the "x" variable are the same:
2(3x+4y=-17) --> 6x+8y=-34
3(2x+3y=-13) --> 6x+9y=-39
Now, do the elimination:
6x+8y=-34
- 6x+9y=-39
-------------------
-y=5 --> y=-5
Now that you found "y", you can just substitute that value in one of the two equations. I will use the original first equation:
3x+4(-5)=-17
3x-20=-17
3x=3 --> x=1
So, we come to the conclusion that x=1, and y=-5.
To solve the system of equations, you can use either the substitution or elimination method. Let's use the elimination method.
The given system of equations is:
(1) 3x + 4y = -17
(2) 2x + 3y = -13
To eliminate one variable, we need to multiply one or both equations by a constant so that the coefficient of either x or y becomes the same in both equations. In this case, let's eliminate y.
Multiply equation (1) by 3 and equation (2) by 4 to make the coefficients of y the same:
(3) 9x + 12y = -51
(4) 8x + 12y = -52
Now, subtract equation (4) from equation (3) to eliminate y:
(3) - (4) gives:
(9x - 8x) + (12y - 12y) = (-51 - (-52))
x + 0 = 1
x = 1
Now substitute the value of x back into either equation (1) or (2) to find the value of y. Let's substitute into equation (1):
3(1) + 4y = -17
3 + 4y = -17
4y = -17 - 3
4y = -20
y = -20/4
y = -5
Therefore, the solution to the system of equations is x = 1 and y = -5.
Hence, the system has a unique solution.
To solve this system of linear equations, you can use either substitution or elimination method. I will explain both methods, and then you can choose which one you prefer to use.
1. Substitution Method:
In the substitution method, we solve one equation for one variable and substitute that expression into the other equation. Here's how to do it step by step:
Step 1: Solve one equation for one variable. We'll solve the first equation for x:
3x + 4y = -17
3x = -17 - 4y
x = (-17 - 4y) / 3
Step 2: Substitute that expression for x into the second equation:
2x + 3y = -13
2((-17 - 4y) / 3) + 3y = -13
Now, we can solve this equation to find the value of y. Once we find y, we can substitute it back into one of the original equations to find the value of x. If the resulting values of x and y satisfy both equations, then the system has a unique solution. If not, it will have either infinitely many solutions or no solution.
2. Elimination Method:
In the elimination method, we modify the equations so that when we add or subtract them, one of the variables cancels out. Here's how to do it step by step:
Step 1: Multiply both sides of the first equation by 2 and the second equation by 3 to make the coefficients of x in each equation the same.
6x + 8y = -34
6x + 9y = -39
Step 2: Subtract the first equation from the second equation to eliminate x:
(6x + 9y) - (6x + 8y) = -39 - (-34)
y = -5
Step 3: Substitute the value of y back into either equation to solve for x:
3x + 4y = -17
3x + 4(-5) = -17
3x - 20 = -17
3x = 3
x = 1
Now, substitute the values of x and y into both equations to check if they are satisfied. If they are, then the system has a unique solution. If not, it will have either infinitely many solutions or no solution.
I hope this explanation helps you solve the system of equations and determine whether it has a unique solution, infinitely many solutions, or no solution. Let me know if you have any further questions!