To solve the system of equations:
[tex]\[
\begin{array}{l}
7 y + 10 x = -11 \\
4 y - 3 x = -15 \\
\end{array}
\][/tex]
we will use either substitution or elimination. Here, I will detail the elimination method.
### Step 1: Align the equations
First, we write down the two given equations as they are:
[tex]\[
7 y + 10 x = -11 \tag{1}
\][/tex]
[tex]\[
4 y - 3 x = -15 \tag{2}
\][/tex]
### Step 2: Eliminate one variable
To eliminate one of the variables, we can create equivalent coefficients for either [tex]\(x\)[/tex] or [tex]\(y\)[/tex]. In this case, let's eliminate [tex]\(y\)[/tex].
Multiply equation (1) by 4 and equation (2) by 7:
[tex]\[
4 \cdot (7 y + 10 x) = 4 \cdot (-11)
\][/tex]
[tex]\[
28 y + 40 x = -44 \tag{3}
\][/tex]
[tex]\[
7 \cdot (4 y - 3 x) = 7 \cdot (-15)
\][/tex]
[tex]\[
28 y - 21 x = -105 \tag{4}
\][/tex]
### Step 3: Subtract the equations
Next, subtract equation (4) from equation (3):
[tex]\[
(28 y + 40 x) - (28 y - 21 x) = -44 - (-105)
\][/tex]
This simplifies to:
[tex]\[
28 y + 40 x - 28 y + 21 x = -44 + 105
\][/tex]
[tex]\[
61 x = 61
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
Divide both sides by 61:
[tex]\[
x = 1
\][/tex]
### Step 5: Substitute back to find [tex]\(y\)[/tex]
Now substitute [tex]\(x = 1\)[/tex] back into one of the original equations. Let's use equation (1):
[tex]\[
7 y + 10 (1) = -11
\][/tex]
[tex]\[
7 y + 10 = -11
\][/tex]
Subtract 10 from both sides:
[tex]\[
7 y = -21
\][/tex]
Divide both sides by 7:
[tex]\[
y = -3
\][/tex]
### Final Answer
Thus, the solution to the system of equations is:
[tex]\[
x = 1
\][/tex]
[tex]\[
y = -3
\][/tex]
