Certainly! Let's solve the system of linear equations step-by-step.
We have two equations:
[tex]\[
8x + 5y = 720
\][/tex]
[tex]\[
10x + 9y = 1098
\][/tex]
### Step 1: Express one of the variables in terms of the other using the first equation.
Let's solve the first equation for [tex]\( y \)[/tex]:
[tex]\[
8x + 5y = 720
\][/tex]
Isolate [tex]\( y \)[/tex]:
[tex]\[
5y = 720 - 8x
\][/tex]
[tex]\[
y = \frac{720 - 8x}{5}
\][/tex]
### Step 2: Substitute this expression for [tex]\( y \)[/tex] into the second equation.
Starting with the second equation:
[tex]\[
10x + 9y = 1098
\][/tex]
Substitute [tex]\( y = \frac{720 - 8x}{5} \)[/tex] into the second equation:
[tex]\[
10x + 9\left(\frac{720 - 8x}{5}\right) = 1098
\][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex].
Multiply through by 5 to clear the fraction:
[tex]\[
50x + 9(720 - 8x) = 5490
\][/tex]
Distribute the 9:
[tex]\[
50x + 6480 - 72x = 5490
\][/tex]
Combine like terms:
[tex]\[
50x - 72x + 6480 = 5490
\][/tex]
[tex]\[
-22x + 6480 = 5490
\][/tex]
Isolate [tex]\( x \)[/tex]:
[tex]\[
-22x = 5490 - 6480
\][/tex]
[tex]\[
-22x = -990
\][/tex]
[tex]\[
x = \frac{-990}{-22}
\][/tex]
[tex]\[
x = 45
\][/tex]
### Step 4: Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex].
Using [tex]\( y = \frac{720 - 8x}{5} \)[/tex]:
[tex]\[
y = \frac{720 - 8(45)}{5}
\][/tex]
[tex]\[
y = \frac{720 - 360}{5}
\][/tex]
[tex]\[
y = \frac{360}{5}
\][/tex]
[tex]\[
y = 72
\][/tex]
### Step 5: Verify the solutions.
Plug [tex]\( x = 45 \)[/tex] and [tex]\( y = 72 \)[/tex] back into the original equations to check:
First equation:
[tex]\[
8(45) + 5(72) = 720
\][/tex]
[tex]\[
360 + 360 = 720
\][/tex]
[tex]\[
720 = 720
\][/tex]
Second equation:
[tex]\[
10(45) + 9(72) = 1098
\][/tex]
[tex]\[
450 + 648 = 1098
\][/tex]
[tex]\[
1098 = 1098
\][/tex]
Both equations are satisfied. Thus, the solutions are:
[tex]\[
x = 45
\][/tex]
[tex]\[
y = 72
\][/tex]
