Question

1. find y.

image of parallel lines cut by two transversals, with segments labeled 9, 21, y, and y + 7

Explanation:

Step1: Identify Proportional Segments

The lines are parallel, so the segments are proportional. Thus, $\frac{9}{21}=\frac{y}{y + 7}$.

Step2: Cross - Multiply

Cross - multiply to get $9(y + 7)=21y$.

Step3: Expand and Simplify

Expand: $9y+63 = 21y$. Subtract $9y$: $63 = 12y$.

Step4: Solve for y

Divide by 12: $y=\frac{63}{12}=\frac{21}{4}=5.25$. Wait, correction: Wait, cross - multiply again. Wait, $\frac{9}{21}=\frac{y}{y + 7}$ simplifies to $\frac{3}{7}=\frac{y}{y + 7}$. Then $3(y + 7)=7y$, $3y+21 = 7y$, $21 = 4y$, $y=\frac{21}{4}=5.25$? Wait, no, wait 9 and 21: GCD of 9 and 21 is 3, so $\frac{9}{21}=\frac{3}{7}$. So $\frac{3}{7}=\frac{y}{y + 7}$. Cross - multiply: $3(y + 7)=7y$, $3y+21 = 7y$, $21 = 4y$, $y=\frac{21}{4}=5.25$? Wait, no, maybe I made a mistake. Wait, 9 and 21: 9/21 = 3/7. So 3/7 = y/(y + 7). So 3(y + 7)=7y → 3y+21 = 7y → 21 = 4y → y = 21/4 = 5.25? Wait, but let's check again. Wait, the two transversals cut the parallel lines, so the ratios of the segments are equal. So the first transversal has segments 9 and 21, the second has y and y + 7. So 9/21 = y/(y + 7). Simplify 9/21 to 3/7. Then 3/7 = y/(y + 7). Cross - multiply: 3(y + 7)=7y → 3y+21 = 7y → 21 = 4y → y = 21/4 = 5.25. Wait, but maybe I mixed up the segments. Wait, maybe the ratio is 9/(9 + 21)=y/(y+(y + 7))? No, no, the segments between the parallel lines on each transversal. So the first transversal has two segments: 9 and 21 (wait, no, maybe 9 is one segment, 21 is another? Wait, looking at the diagram, the first transversal (the non - slanted? No, the slanted transversals. Wait, the two slanted lines are transversals, cutting the three parallel lines. So on the first transversal, the lengths are 9 and 21 (between the first and second parallel line, and second and third? Wait, maybe the segments are 9 (between top and middle) and 21 (between middle and bottom) on one transversal, and y (top - middle) and y + 7 (middle - bottom) on the other. So the ratio of top - middle to middle - bottom on each transversal is equal. So 9/21 = y/(y + 7). Yes, that's correct. So 9/21 = y/(y + 7) → 3/7 = y/(y + 7) → 3(y + 7)=7y → 3y+21 = 7y → 21 = 4y → y = 21/4 = 5.25. Wait, but 21/4 is 5.25. Let's check: y = 21/4, y + 7=21/4+28/4 = 49/4. Then y/(y + 7)=(21/4)/(49/4)=21/49 = 3/7, and 9/21 = 3/7. So it works. So y = 21/4 = 5.25 or 5.25.

Answer:

$\frac{21}{4}$ (or 5.25)