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Changing the Subject of a Formula - Questions

Question 1
49303

Make \(y\) the subject of \(x=y^{2}+1\)

\(y=\pm \sqrt{x-1}\)

\(\begin{aligned}
x&=y^{2}+1 \\
y^{2}+1&=x \\
y^{2}+1-1&=x-1 \\
y^{2}&=x-1 \\
y&=\pm \sqrt{x-1}
\end{aligned}\)

Question 2
49304

Make \(r\) the subject of \(A=\pi r^{2}\quad (r>0)\)

\(r=\sqrt{\dfrac{A}{\pi}}\) 

\(\begin{aligned}
A&=\pi r^{2} \quad(r>0) \\
\pi r^{2}&=A \\
\frac{\pi r^{2}}{\pi}&=\frac{A}{\pi} \\
r^{2}&=\frac{A}{\pi} \\
r&=\sqrt{\frac{A}{\pi}}
\end{aligned}\)

Question 3
49309

Make \(a\) the subject of \(\dfrac{a}{4}=ax+y\)

\(a=\dfrac{4y}{1-4 x}\)  

\(\begin{aligned}
 \frac{a}{4}&=ax+y \\
4 \times \frac{a}{4}&=4(a x+y) \\
a&=4ax+4y  \\
4ax+4y&=a \\
4ax+4y-4y&=a-4y \\
4ax&=a-4y \\
4ax&=-4y+a \\
4ax-a&=-4y+a-a\\
4ax-a&=-4y \\
a(4x-1)&=-4y \\
a&=\frac{-4y}{4x-1} \\
a&=\frac{4y}{1-4x}
\end{aligned}\)

Question 4
49310

Make \(a\) the subject of \(b=\dfrac{a+1}{a-1}\)

\(a =\dfrac{b+1}{b-1}\) 

\(\begin{aligned}
b &=\frac{a+1}{a-1} \\
\frac{a+1}{a-1} &=b \\
a+1 &=b(a-1) \\
a+1 &=ba-b \\
a &=b a-b-1 \\
a-ba &=-b-1 \\
ba-a &=b+1 \\
a(b-1) &=b+1 \\
a &=\frac{b+1}{b-1}
\end{aligned}\)

Question 5
49305

Make \(h\) the subject of \(A=\dfrac{1}{2} bh\)

\(h =\dfrac{2A}{b}\) 

\(\begin{aligned}
A &=\frac{1}{2} b h \\
\frac{1}{2} b h &=A \\
2 \times \frac{1}{2} b h &=2 \times A \\
bh &=2 A \\
\frac{bh}{b} &=\frac{2A}{b} \\
h &=\frac{2A}{b}
\end{aligned}\)

Question 6
49306

Make \(x\) the subject of \(ax+by+c=0\)

\(x=\dfrac{-c-by}{a}\)  

\( \begin{aligned}
ax+by+c&=0 \\
ax+by+c-c&=0-c \\
ax+by&=-c \\
ax+by-by&=-c-b y \\
ax&=-c-by \\
\frac{ax}{a}&=\frac{-c-by}{a} \\
x&=\frac{-c-by}{a}
\end{aligned}\)

Question 7
49307

Make \(a\) the subject of \(v=u+at\)

\(a=\dfrac{v-u}{t}\) 

\(\begin{aligned}
v&=u+at \\
u+at&=v \\
at+u&=v \\
at+u-u&=v-u \\
at&=v-u \\
\frac{at}{t}&=\frac{v-u}{t} \\
a&=\frac{v-u}{t}
\end{aligned}\)

Question 8
49308

Make \(a\) the subject of \(\dfrac{\sin A}{a}=\dfrac{\sin B}{b}\)

\(a =\dfrac{b \sin A}{\sin B}\)

\(\begin{aligned}
\frac{\sin A}{a} &=\frac{\sin B}{b} \\
\frac{a}{\sin A} &=\frac{b}{\sin B} \\
\frac{a}{\sin A} \times \sin A &=\frac{b}{\sin B} \times \sin A \\
a &=\frac{b \sin A}{\sin B}
\end{aligned}\)

Question 9
15538

Change the subject of the formula to \(y \to \,\,m = \dfrac{{2xy}}{{x + y}}\) 

\(y = \dfrac{{mx}}{{2x - m}}\)

Question 10
15535

Change the subject of the formula to \(r \to \,\,V = \dfrac{2}{3}\;\pi {r^3}\) 

\(r = \sqrt[3]{{\dfrac{{3V}}{{2\pi }}}}\)

Question 11
15537

Change the subject of the formula to \(F \to \,\,C = \dfrac{5}{9}\left( {F - 32} \right)\)

\(F = \dfrac{{9C}}{5} + 32\)

Question 12
15532

Change the subject of the formula to \(p \to \,\,q = \dfrac{p}{m} + n\)

\(p = m(q - n)\)

Question 13
15534

Change the subject of the formula to \(y \to \,\,x = \dfrac{m}{{\sqrt y }}\) 

\(y = \dfrac{{{m^2}}}{{{x^2}}}\)