Answer :

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]