To solve this system of equations:

1. $5x + 6y = -6$
2. $7x - 3y = -12$

Let's use the elimination method to find the values of $x$ and $y$.

Step 1: Make the Coefficients of $y$ Equal

To eliminate $y$, we can multiply the first equation by 3 and the second equation by 6:

$3(5x + 6y) = 3(-6) \Rightarrow 15x + 18y = -18$ $6(7x - 3y) = 6(-12) \Rightarrow 42x - 18y = -72$

Now we have:

1. $15x + 18y = -18$
2. $42x - 18y = -72$

Step 2: Add the Equations

Add the two equations to eliminate $y$:

$(15x + 18y) + (42x - 18y) = -18 - 72$ $57x = -90$

Step 3: Solve for $x$

Divide both sides by 57:

$x = \frac{-90}{57} = -\frac{30}{19}$

Step 4: Substitute $x$ Back into One of the Original Equations

Now, substitute $x = -\frac{30}{19}$ into the first equation to solve for $y$:

$5\left(-\frac{30}{19}\right) + 6y = -6$ $-\frac{150}{19} + 6y = -6$

Convert -6 to a fraction with a denominator of 19:

$-\frac{150}{19} + 6y = -\frac{114}{19}$

Add $\frac{150}{19}$ to both sides:

$6y = \frac{36}{19}$

Now, divide by 6:

$y = \frac{36}{19} \times \frac{1}{6} = \frac{6}{19}$

Solution

The solution to the system is: $x = -\frac{30}{19}, \quad y = \frac{6}{19}$

Would you like further details on each step or have any questions?

Here are some related questions to explore further:

1. How can we solve this system of equations using substitution?
2. What is the significance of using fractions instead of decimal approximations in exact solutions?
3. Can we solve this system by graphing, and what does the solution represent graphically?
4. How does the elimination method compare to substitution in terms of efficiency?
5. What strategies can we use if both elimination and substitution methods seem challenging?

Tip: Always check your solution by plugging the values back into the original equations to ensure accuracy.