Question

Solve the following system of equations, state the method that you would use, and show all work to solve:

5x - 3y = - 4
3x - y = - 4

x + 5y = 12
5x + 25y = 12

x + y = - 2
- 3x - 3y = 6

Answer 1

Answer on Question#43683 – Math – Abstract Algebra

Question:

Solve the following system of equations, state the method that you would use, and show all work to solve:

$$\begin{array}{l} 5x - 3y = -4 \\ 3x - y = -4 \\ x + 5y = 12 \\ 5x + 25y = 12 \\ x + y = -2 \\ -3x - 3y = 6 \\ \end{array}$$

Solution.

To solve the first system we will use the substitution method.

From second equation we have: $y = 3x + 4$. Substituting it in the first equation we get:

$$\begin{array}{l} 5x - 3(3x + 4) = -4 \\ 5x - 9x - 12 = -4 \\ -4x = 8 \\ x = -2. \\ \end{array}$$

And coming back to the first equation we get:

$y = 3(-2) + 4 = -2$. So, the solution is $x = -2$, $y = -2$.

Answer. $x = -2, y = -2$.

To solve the second equation let's use the substitution method again.

From first equation we have: $x = 12 - 5y$. Substituting it in the second equation we get

$$\begin{array}{l} 5(12 - 5y) + 25y = 12 \\ 60 - 25y + 25y = 12 \\ \end{array}$$

we get $60 = 12$

so, we see that this system has no solution.

Answer. No solution.

And to solve the last equation we will use the addition method.

Firstly let's divide the second equation by -3, we get: $x+y=-2$. Hence, the system is consisting from 2 identical equations. Hence, we get the whole line of solutions $y=-2-x$.

Answer. The solutions are all points which satisfies $y = -2 - x$.

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