Question

directions: solve the equation for the specified variable and circle your answer to indicate the color. then, evaluate the equation for the given replacement values to determine which number gets that color.
a solve ( a=\frac{b}{c^{2}} ) for ( b ). ( b=\frac{a}{c^{2}} ) light blue ( b=\frac{c^{2}}{a} ) orange ( b=a c^{2} ) dark green if ( a=\frac{1}{8} ) and ( c = 4 ), find ( b ).
b solve ( m - v = k ) for ( v ). ( v = k + m ) brown ( v = m - k ) light blue ( v = k - m ) yellow if ( k=-7 ) and ( m=-2 ), find ( v ).
c solve ( \frac{p}{q}=\frac{r}{s} ) for ( q ). ( q = p s r ) orange ( q=\frac{p s}{r} ) yellow ( q=\frac{r}{p s} ) purple if ( p = 14, r=-7 ), and ( s=-4 ), find ( q ).
d solve ( 4 m+2 n=5 n ) for ( m ). ( m=-\frac{3 n}{4} ) dark blue ( m=\frac{4 n}{3} ) purple ( m=\frac{3 n}{4} ) red if ( n = 8 ), find ( m ).
e solve ( w=\frac{x}{y}-z ) for ( x ). ( x = y(w + z) ) light green ( x = w y+z ) brown ( x=\frac{y}{w + z} ) orange if ( w=-8, y=-5 ), and ( z = 6 ), find ( x ).

Explanation:

Step1: Solve \(a=\frac{b}{c^{2}}\) for \(b\)

Multiply both sides by \(c^{2}\): \(a\times c^{2}=\frac{b}{c^{2}}\times c^{2}\), so \(b = ac^{2}\).

Step2: Evaluate \(b\) when \(a=\frac{1}{8}\) and \(c = 4\)

Substitute \(a=\frac{1}{8}\) and \(c = 4\) into \(b = ac^{2}\). First, calculate \(c^{2}=4^{2}=16\). Then \(b=\frac{1}{8}\times16\).
\(b = 2\).

Step3: Solve \(m - v=k\) for \(v\)

Subtract \(m\) from both sides: \(-v=k - m\). Multiply both sides by \(- 1\): \(v=m - k\).

Step4: Evaluate \(v\) when \(k=-7\) and \(m=-2\)

Substitute \(k = - 7\) and \(m=-2\) into \(v=m - k\). \(v=-2-(-7)=-2 + 7=5\).

Step5: Solve \(\frac{p}{q}=\frac{r}{s}\) for \(q\)

Cross - multiply: \(p\times s=r\times q\). Then \(q=\frac{ps}{r}\).

Step6: Evaluate \(q\) when \(p = 14\), \(r=-7\), and \(s=-4\)

Substitute \(p = 14\), \(r=-7\), and \(s=-4\) into \(q=\frac{ps}{r}\). \(ps=14\times(-4)=-56\). Then \(q=\frac{-56}{-7}=8\).

Step7: Solve \(4m+2n = 5n\) for \(m\)

Subtract \(2n\) from both sides: \(4m=5n-2n\). So \(4m = 3n\). Divide both sides by \(4\): \(m=\frac{3n}{4}\).

Step8: Evaluate \(m\) when \(n = 8\)

Substitute \(n = 8\) into \(m=\frac{3n}{4}\). \(m=\frac{3\times8}{4}=6\).

Step9: Solve \(w=\frac{x}{y}-z\) for \(x\)

Add \(z\) to both sides: \(w + z=\frac{x}{y}\). Multiply both sides by \(y\): \(x=y(w + z)\).

Step10: Evaluate \(x\) when \(w=-8\), \(y=-5\), and \(z = 6\)

Substitute \(w=-8\), \(y=-5\), and \(z = 6\) into \(x=y(w + z)\). \(w + z=-8 + 6=-2\). Then \(x=-5\times(-2)=10\).

Answer:

For A: \(b = ac^{2}\), \(b = 2\); For B: \(v=m - k\), \(v = 5\); For C: \(q=\frac{ps}{r}\), \(q = 8\); For D: \(m=\frac{3n}{4}\), \(m = 6\); For E: \(x=y(w + z)\), \(x = 10\)