Sure, let's go through the steps together to determine the correct equations in standard form.
Initially, we are given the following equations based on the weights:
1. [tex]\(\frac{1}{2} t + \frac{3}{4} s = 43 \frac{3}{4}\)[/tex]
2. [tex]\(\frac{3}{4} s = -\frac{1}{2} t + 43 \frac{3}{4}\)[/tex]
3. [tex]\(2 t + 3 s = 175\)[/tex]
4. [tex]\(3 s = -2 t + 175\)[/tex]
### Step 1: Convert each equation to standard form (if necessary)
Equation 1:
[tex]\[\frac{1}{2} t + \frac{3}{4} s = 43 \frac{3}{4}\][/tex]
- Multiply through by 4 to clear the fractions:
[tex]\[4 \left(\frac{1}{2} t + \frac{3}{4} s\right) = 4 \times 43 \frac{3}{4}\][/tex]
[tex]\[2 t + 3 s = 175\][/tex]
Equation 2:
[tex]\[\frac{3}{4} s = -\frac{1}{2} t + 43 \frac{3}{4}\][/tex]
- Multiply through by 4 to clear the fractions:
[tex]\[4 \left(\frac{3}{4} s\right) = 4 \left(-\frac{1}{2} t + 43 \frac{3}{4}\right)\][/tex]
[tex]\[3 s = -2 t + 175\][/tex]
### Step 2: Verify the standard form of the equations
Equation 3:
[tex]\[2 t + 3 s = 175\][/tex]
- We notice that this is already provided in standard form.
Equation 4:
[tex]\[3 s = -2 t + 175\][/tex]
- Similarly, this is already in standard form.
### Conclusion:
After converting the given equations into standard forms, we see that:
[tex]\[
2t + 3s = 175
\][/tex]
is a valid standard form equation representing the relationship between the number of T-shirts [tex]\( t \)[/tex] and sweatshirts [tex]\( s \)[/tex] ordered by the students.
Therefore, the equation [tex]\( 2 t + 3 s = 175 \)[/tex] represents the number of T-shirts [tex]\( t \)[/tex] and the number of sweatshirts [tex]\( s \)[/tex] that were ordered.
