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Answer :
We start with the expression
[tex]$$30\left(\frac{1}{2} x-2\right)+40\left(\frac{3}{4} y-4\right).$$[/tex]
Step 1. Distribute in the first term:
Multiply [tex]$30$[/tex] by both terms inside the parentheses:
- Multiply [tex]$30$[/tex] by [tex]$\frac{1}{2} x$[/tex]:
[tex]$$30 \cdot \frac{1}{2} x = 15x.$$[/tex]
- Multiply [tex]$30$[/tex] by [tex]$-2$[/tex]:
[tex]$$30 \cdot (-2) = -60.$$[/tex]
So, the first term becomes:
[tex]$$15x - 60.$$[/tex]
Step 2. Distribute in the second term:
Multiply [tex]$40$[/tex] by both terms inside the parentheses:
- Multiply [tex]$40$[/tex] by [tex]$\frac{3}{4} y$[/tex]:
[tex]$$40 \cdot \frac{3}{4} y = 30y.$$[/tex]
- Multiply [tex]$40$[/tex] by [tex]$-4$[/tex]:
[tex]$$40 \cdot (-4) = -160.$$[/tex]
So, the second term becomes:
[tex]$$30y - 160.$$[/tex]
Step 3. Combine the distributed terms:
Now add the results from the two steps:
[tex]$$15x - 60 + 30y - 160.$$[/tex]
Combine the constant terms ([tex]$-60$[/tex] and [tex]$-160$[/tex]):
[tex]$$-60 - 160 = -220.$$[/tex]
Thus, the simplified expression is:
[tex]$$15x + 30y - 220.$$[/tex]
The equivalent expression is:
[tex]$$\boxed{15x + 30y - 220}.$$[/tex]
[tex]$$30\left(\frac{1}{2} x-2\right)+40\left(\frac{3}{4} y-4\right).$$[/tex]
Step 1. Distribute in the first term:
Multiply [tex]$30$[/tex] by both terms inside the parentheses:
- Multiply [tex]$30$[/tex] by [tex]$\frac{1}{2} x$[/tex]:
[tex]$$30 \cdot \frac{1}{2} x = 15x.$$[/tex]
- Multiply [tex]$30$[/tex] by [tex]$-2$[/tex]:
[tex]$$30 \cdot (-2) = -60.$$[/tex]
So, the first term becomes:
[tex]$$15x - 60.$$[/tex]
Step 2. Distribute in the second term:
Multiply [tex]$40$[/tex] by both terms inside the parentheses:
- Multiply [tex]$40$[/tex] by [tex]$\frac{3}{4} y$[/tex]:
[tex]$$40 \cdot \frac{3}{4} y = 30y.$$[/tex]
- Multiply [tex]$40$[/tex] by [tex]$-4$[/tex]:
[tex]$$40 \cdot (-4) = -160.$$[/tex]
So, the second term becomes:
[tex]$$30y - 160.$$[/tex]
Step 3. Combine the distributed terms:
Now add the results from the two steps:
[tex]$$15x - 60 + 30y - 160.$$[/tex]
Combine the constant terms ([tex]$-60$[/tex] and [tex]$-160$[/tex]):
[tex]$$-60 - 160 = -220.$$[/tex]
Thus, the simplified expression is:
[tex]$$15x + 30y - 220.$$[/tex]
The equivalent expression is:
[tex]$$\boxed{15x + 30y - 220}.$$[/tex]
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