Question

equation puzzle 6
name
use your equation solving skills to find the value of each variable. use the values you calculate to help you move along in the puzzle and find all the missing variables.
$-8\left(2k + \frac{3}{4}\
ight) + 5 = -121$ \t $k = 7.5$
$\frac{\frac{1}{2}(8k + l)}{6} = 4\frac{2}{3}$ \t $l = -4$
$\frac{1}{5}(4m + l) - \frac{1}{2} = -5\frac{1}{2}$ \t $m = $
$\frac{10(n - m)}{\sfrac{1}{2}} + 1\frac{1}{2} = 109$ \t $n = $
$n(10p - 16) + 3\frac{1}{2} = \frac{1}{2}$ \t $p = $

Explanation:

Step1: Solve for \( l \) (Verify or Solve)

We know \( k = 7.5=\frac{15}{2} \), and the equation is \( \frac{\frac{1}{2}(8k + l)}{6}=4\frac{2}{3}=\frac{14}{3} \)
Multiply both sides by 6: \( \frac{1}{2}(8k + l)=6\times\frac{14}{3} = 28 \)
Multiply both sides by 2: \( 8k + l = 56 \)
Substitute \( k=\frac{15}{2} \): \( 8\times\frac{15}{2}+l = 56 \)
\( 60 + l = 56 \)
\( l=56 - 60=-4 \) (matches the given \( l = - 4\))

Step2: Solve for \( m \)

Equation: \( \frac{1}{5}(4m + l)-\frac{1}{2}=-5\frac{1}{2}=-\frac{11}{2} \)
Add \( \frac{1}{2} \) to both sides: \( \frac{1}{5}(4m + l)=-\frac{11}{2}+\frac{1}{2}=-5 \)
Multiply both sides by 5: \( 4m + l=-25 \)
Substitute \( l = - 4 \): \( 4m-4=-25 \)
Add 4 to both sides: \( 4m=-21 \)
Divide by 4: \( m =-\frac{21}{4}=-5.25 \)

Step3: Solve for \( n \)

Equation: \( \frac{10(n - m)}{\frac{1}{2}}+1\frac{1}{2}=109 \)
Subtract \( 1\frac{1}{2}=\frac{3}{2} \) from both sides: \( \frac{10(n - m)}{\frac{1}{2}}=109-\frac{3}{2}=\frac{218 - 3}{2}=\frac{215}{2} \)
Multiply both sides by \( \frac{1}{2} \): \( 10(n - m)=\frac{215}{2}\times\frac{1}{2}=\frac{215}{4} \)? Wait, no, \( \frac{10(n - m)}{\frac{1}{2}}=10(n - m)\times2 = 20(n - m) \)
So correct step: \( 20(n - m)=109-\frac{3}{2}=\frac{218 - 3}{2}=\frac{215}{2} \)? No, \( 109-\frac{3}{2}=\frac{218 - 3}{2}=\frac{215}{2} \) is wrong. \( 109=\frac{218}{2} \), so \( \frac{218}{2}-\frac{3}{2}=\frac{215}{2} \). But \( 20(n - m)=\frac{215}{2} \)? Wait, no, \( \frac{10(n - m)}{\frac{1}{2}}=10(n - m)\div\frac{1}{2}=10(n - m)\times2 = 20(n - m) \)
So \( 20(n - m)=109 - \frac{3}{2}=\frac{218 - 3}{2}=\frac{215}{2} \)? No, \( 109=\frac{218}{2} \), so \( \frac{218}{2}-\frac{3}{2}=\frac{215}{2} \). Then \( n - m=\frac{215}{2}\div20=\frac{215}{40}=\frac{43}{8} \)? Wait, no, earlier \( m =-\frac{21}{4}=-5.25 \)
Wait, let's redo:
\( \frac{10(n - m)}{\frac{1}{2}}=10(n - m)\times2 = 20(n - m) \)
Equation: \( 20(n - m)+\frac{3}{2}=109 \)
Subtract \( \frac{3}{2} \): \( 20(n - m)=109-\frac{3}{2}=\frac{218 - 3}{2}=\frac{215}{2} \)? No, \( 109=\frac{218}{2} \), so \( \frac{218}{2}-\frac{3}{2}=\frac{215}{2} \). Then \( n - m=\frac{215}{2}\div20=\frac{215}{40}=\frac{43}{8}=5.375 \)
But \( m =-\frac{21}{4}=-5.25 \)
So \( n=5.375 + (-5.25)=0.125=\frac{1}{8} \)? Wait, no, mistake in step. Wait, original equation: \( \frac{10(n - m)}{\frac{1}{2}}+1\frac{1}{2}=109 \)
\( \frac{10(n - m)}{\frac{1}{2}}=10(n - m)\times2 = 20(n - m) \)
So \( 20(n - m)=109 - \frac{3}{2}=\frac{218 - 3}{2}=\frac{215}{2} \)? No, \( 109=\frac{218}{2} \), \( \frac{218}{2}-\frac{3}{2}=\frac{215}{2} \). Then \( n - m=\frac{215}{2}\div20=\frac{215}{40}=\frac{43}{8} \). But \( m =-\frac{21}{4}=-5.25=-\frac{21}{4}=-\frac{42}{8} \)
So \( n=\frac{43}{8}-\frac{42}{8}=\frac{1}{8} \)? No, that can't be. Wait, maybe I messed up \( m \)'s calculation.
Wait, step 2: \( \frac{1}{5}(4m + l)-\frac{1}{2}=-\frac{11}{2} \)
Add \( \frac{1}{2} \): \( \frac{1}{5}(4m + l)=-\frac{11}{2}+\frac{1}{2}=-5 \)
Multiply by 5: \( 4m + l=-25 \), \( l=-4 \), so \( 4m-4=-25 \), \( 4m=-21 \), \( m =-\frac{21}{4}=-5.25 \) (correct)
Now step 3: \( \frac{10(n - m)}{\frac{1}{2}}+1\frac{1}{2}=109 \)
\( \frac{10(n - m)}{\frac{1}{2}}=10(n - m)\times2 = 20(n - m) \)
So \( 20(n - m)=109 - \frac{3}{2}=\frac{218 - 3}{2}=\frac{215}{2} \)? No, \( 109=\frac{218}{2} \), \( \frac{218}{2}-\frac{3}{2}=\frac{215}{2} \). Then \( 20(n - m)=\frac{215}{2} \), \( n - m=\frac{215}{2}\div20=\frac{215}{40}=\frac{43}{8}=5.375 \)
\( m = - 5.25=-\frac{21}{4}=-5.25 \), so \( n=5.375+(-5.25)=0.125=\frac{1}{8} \)? Wait, but let's check with another approach. Wait, maybe the equation is \( \frac{10(n - m)}{\frac{1}{2}}=10(n - m)\times2 = 20(n - m) \), so \( 20(n - m)=109 - 1.5 = 107.5 \)
\( 20(n - m)=107.5 \), \( n - m=107.5\div20 = 5.375 \)
\( m=-5.25 \), so \( n=5.375-5.25 = 0.125=\frac{1}{8} \)? No, that seems odd. Wait, maybe I made a mistake in the equation's interpretation. The equation is \( \frac{10(n - m)}{\frac{1}{2}}+1\frac{1}{2}=109 \)
\( \frac{10(n - m)}{\frac{1}{2}}=10(n - m)\times2 = 20(n - m) \), so \( 20(n - m)=109 - 1.5 = 107.5 \)
\( n - m=107.5\div20 = 5.375 \)
\( m=-5.25 \), so \( n=5.375 + (-5.25)=0.125=\frac{1}{8} \) (but let's check step 4)

Step4: Solve for \( p \)

Equation: \( n(10p - 16)+3\frac{1}{2}=\frac{1}{2} \)
Subtract \( 3\frac{1}{2}=\frac{7}{2} \): \( n(10p - 16)=\frac{1}{2}-\frac{7}{2}=-3 \)
Substitute \( n=\frac{1}{8} \): \( \frac{1}{8}(10p - 16)=-3 \)
Multiply by 8: \( 10p - 16=-24 \)
Add 16: \( 10p=-8 \)
Divide by 10: \( p =-\frac{4}{5}=-0.8 \)

Wait, but maybe I made a mistake in \( n \)'s calculation. Let's re - solve \( n \):

Equation for \( n \): \( \frac{10(n - m)}{\frac{1}{2}}+1\frac{1}{2}=109 \)
\( \frac{10(n - m)}{\frac{1}{2}}=10(n - m)\times2 = 20(n - m) \)
\( 20(n - m)=109 - \frac{3}{2}=\frac{218 - 3}{2}=\frac{215}{2} \)? No, \( 109=\frac{218}{2} \), \( \frac{218}{2}-\frac{3}{2}=\frac{215}{2} \). Then \( n - m=\frac{215}{2}\div20=\frac{215}{40}=\frac{43}{8} \). \( m =-\frac{21}{4}=-\frac{42}{8} \), so \( n=\frac{43}{8}-\frac{42}{8}=\frac{1}{8} \) (correct). Then for \( p \):

\( n=\frac{1}{8} \), so \( \frac{1}{8}(10p - 16)=-3 \)
\( 10p-16=-24 \)
\( 10p=-8 \)
\( p =-\frac{4}{5}=-0.8 \)

Answer:

\( m =-\frac{21}{4}=-5.25 \), \( n=\frac{1}{8}=0.125 \), \( p =-\frac{4}{5}=-0.8 \)

(If we need to present as per the puzzle, for \( m \): \(-5.25\) or \(-\frac{21}{4}\), \( n\): \(0.125\) or \(\frac{1}{8}\), \( p\): \(-0.8\) or \(-\frac{4}{5}\))