Sure, let's solve this step-by-step.
We are given the following system of linear equations:
[tex]\[
\begin{array}{l}
l + p = 9 \\
0.5l + 1.5p = 9.5
\end{array}
\][/tex]
Here, [tex]\( l \)[/tex] represents the number of pounds of limes and [tex]\( p \)[/tex] represents the number of pounds of pears.
### Step 1: Solve for [tex]\( l \)[/tex] in terms of [tex]\( p \)[/tex] using the first equation
From the first equation:
[tex]\[
l + p = 9
\][/tex]
we can isolate [tex]\( l \)[/tex]:
[tex]\[
l = 9 - p
\][/tex]
### Step 2: Substitute [tex]\( l = 9 - p \)[/tex] into the second equation
Now, substitute [tex]\( l = 9 - p \)[/tex] into the second equation:
[tex]\[
0.5l + 1.5p = 9.5
\][/tex]
[tex]\[
0.5(9 - p) + 1.5p = 9.5
\][/tex]
### Step 3: Simplify the equation
Simplify the left-hand side:
[tex]\[
0.5 \cdot 9 - 0.5p + 1.5p = 9.5
\][/tex]
[tex]\[
4.5 - 0.5p + 1.5p = 9.5
\][/tex]
Combine like terms:
[tex]\[
4.5 + p = 9.5
\][/tex]
### Step 4: Solve for [tex]\( p \)[/tex]
Subtract 4.5 from both sides:
[tex]\[
p = 9.5 - 4.5
\][/tex]
[tex]\[
p = 5
\][/tex]
So, Mariah bought 5 pounds of pears.
### Step 5: Solve for [tex]\( l \)[/tex]
Now, substitute [tex]\( p = 5 \)[/tex] back into the equation [tex]\( l = 9 - p \)[/tex]:
[tex]\[
l = 9 - 5
\][/tex]
[tex]\[
l = 4
\][/tex]
So, Mariah bought 4 pounds of limes.
### Conclusion
Mariah bought 4 pounds of limes and 5 pounds of pears.
So, the final answer is:
She bought [tex]\( 4 \)[/tex] pounds of limes and [tex]\( 5 \)[/tex] pounds of pears.
