Answer :
To rewrite the rational expression \(\frac{2}{19x - 304}\) with the denominator \(19y(x - 16)\), follow these steps:
1. Identify the Original Denominator:
The original denominator is \(19x - 304\).
2. Rewrite the Original Denominator:
Notice that \(19x - 304\) can be written as \(19(x - 16)\). This gives us:
[tex]\[ \frac{2}{19x - 304} = \frac{2}{19(x - 16)} \][/tex]
3. Compare the Denominators:
We need to rewrite the rational expression such that it has the denominator \(19y(x - 16)\). The new denominator is \(19y(x - 16)\).
4. Adjust the Numerator:
Since \(19(x - 16)\) is being multiplied by \(y\) to get \(19y(x - 16)\), the numerator must be adjusted accordingly. We must multiply the numerator by 1 (as multiplying by 1 does not change its value):
[tex]\[ \frac{2}{19(x - 16)} = \frac{2 \cdot 1}{19 \cdot 1 \cdot (x - 16)} \][/tex]
5. Final Expression:
To have the denominator \(19y(x - 16)\), our adjusted numerator should still match the original rational expression over the new denominator. Since we are multiplying the denominator by \(y\), we need to ensure the numerator appropriately reflects any changes. The simplest form without further change needed (no additional multiplier necessary given the proportional relationship of 19 to the new denominator):
[tex]\[ \frac{2}{19(x - 16)} = \frac{2 \cdot \frac{1}{y}}{(19(x - 16))} = \frac{2}{19y(x - 16)} \][/tex]
Based on all this, our equivalent rational expression with the new denominator \(19y(x - 16)\) is:
[tex]\[ \frac{2}{19x - 304} = \frac{2}{19y(x - 16)} \][/tex]
So, the final equivalent rational expression is:
[tex]\[ \boxed{\left(2, 19 \right)} \][/tex]
1. Identify the Original Denominator:
The original denominator is \(19x - 304\).
2. Rewrite the Original Denominator:
Notice that \(19x - 304\) can be written as \(19(x - 16)\). This gives us:
[tex]\[ \frac{2}{19x - 304} = \frac{2}{19(x - 16)} \][/tex]
3. Compare the Denominators:
We need to rewrite the rational expression such that it has the denominator \(19y(x - 16)\). The new denominator is \(19y(x - 16)\).
4. Adjust the Numerator:
Since \(19(x - 16)\) is being multiplied by \(y\) to get \(19y(x - 16)\), the numerator must be adjusted accordingly. We must multiply the numerator by 1 (as multiplying by 1 does not change its value):
[tex]\[ \frac{2}{19(x - 16)} = \frac{2 \cdot 1}{19 \cdot 1 \cdot (x - 16)} \][/tex]
5. Final Expression:
To have the denominator \(19y(x - 16)\), our adjusted numerator should still match the original rational expression over the new denominator. Since we are multiplying the denominator by \(y\), we need to ensure the numerator appropriately reflects any changes. The simplest form without further change needed (no additional multiplier necessary given the proportional relationship of 19 to the new denominator):
[tex]\[ \frac{2}{19(x - 16)} = \frac{2 \cdot \frac{1}{y}}{(19(x - 16))} = \frac{2}{19y(x - 16)} \][/tex]
Based on all this, our equivalent rational expression with the new denominator \(19y(x - 16)\) is:
[tex]\[ \frac{2}{19x - 304} = \frac{2}{19y(x - 16)} \][/tex]
So, the final equivalent rational expression is:
[tex]\[ \boxed{\left(2, 19 \right)} \][/tex]
