worksheet 1.2

1. Solve each of the following for $x$

   1. $ax+n=m$
      $ax=m-n$
      $x=\frac{m-n}{a}$

   2. $ax+b=bx$
      $ax-bx=-b$
      $x=\frac{-b}{a-b}$
      $x=\frac{b}{b-a}$

   3. $\frac{ax}{b}+c=0$
      $ax=-cb$
      $x=\frac{-cb}{a}$

   4. $px=qx+5$
      $x(p-q)=5$
      $x=\frac{5}{p-q}$

   5. $mx+n=nx-m$
      $mx-nx=-m-n$
      $x(m-n)=-m-n$
      $x=\frac{-m-n}{m-n}$
      $x=\frac{m+n}{n-m}$

   6. $\frac{1}{x+a}=\frac{b}{x}$
      $x=b(x+a)$
      $x=bx+ba$
      $x-bx=ba$
      $x=\frac{ba}{1-b}$

   7. $\frac{b}{x-a}=\frac{2b}{x+a}$
      $b(x+a)=2b(x-a)$
      $bx+ba=2bx-2ba$
      $2bx-bx=ba+2ba$
      $x=\frac{ba+2ba}{b}$
      $x=3a$

   8. $\frac{x}{m}+n=\frac{x}{n}+m$
      $xn+m{n}^2=xm+m^{2}n$
      $xn-xm=m^{2}n-m{n}^2$
      $x=\frac{m^{2}n-m{n}^2}{n-m}$
      $x=\frac{mn(m-n)}{n-m}$
      $x=\frac{-mn(n-m)}{n-m}$
      $x=-mn$

   9. $-b(ax+b)=a(bx-a)$
      $-bax-b^2=abx-a^2$
      $2abx=a^2-b^2$
      $x=\frac{a^2-b^2}{2ab}$

   10. $p^2(1-x)-2pqx=q^2(1+x)$
       $p^2-x{p}^2-2pqx=q^2+x{q}^2$
       $x{q}^2+2pqx+x{p}^2=p^2-q^2$
       $(x)(p+q{)}^2=(p+q)(p-q)$
       $x=\frac{p-q}{p+q}$

   11. $\frac{x}{a}-1=\frac{x}{b}+2$
       $bx-ab=ax+2ab$
       $bx-ax=3ab$
       $x=\frac{3ab}{b-a}$

   12. $\frac{x}{a-b}+\frac{2x}{a+b}=\frac{1}{a^2-b^2}$
       $xa+xb+2x(a-b)=\frac{(a+b)(a-b)}{(a+b)(a-b)}$
       $x(a+b+2a-2b)=1$
       $x=\frac{1}{3a-b}$

   13. $\frac{p-qx}{t}+p=\frac{qx-t}{p}$
       $p^2-pqx+p^{2}t=qtx-t^2$
       $qtx+pqx=p^{2}t+q^2+t^2$
       $x(qt+pq)=\frac{p^{2}t+q^2+t^2}{qt+pq}$

   14. $\frac{1}{x+a}+\frac{1}{x+2a}=\frac{2}{x+3a}$
       $(x+2a)(x+3a)+(x+a)(x+3a)=2(x+a)(x+2a)$
       $x^2+5ax+6a^2+x^2+4ax+3a^2=2x^2+6ax+4a^2$
       $5ax+6a^2+4ax+3a^2=6ax+4a^2$
       $3ax=-5a^2$
       $x=\frac{-5a}{3}$

2. Consider the simultaneous equations $ax+by=p$ and $bx-ay=q$
   1. Multiply the first equation by a and the second equation by b. Then eliminate y to solve for x.
      $a^{2}x+aby=ap$
      $b^{2}x-aby=bq$
      $x=\frac{ap+bq}{a^2+b^2}$
   2. Multiply the first equation by b and the second equation by -a. Then eliminate x to solve for y.
      $abx+b^{2}y=bp$
      $-abx+a^{2}y=aq$
      $y=\frac{bp+aq}{a^2+b^2}$
3. Consider the simultaneous equations $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$
   1. Multiply the first equation by b and the second equation by -a. Then eliminate y to solve for x.
      $\frac{bx}{a}+y=1$
      $\frac{-ax}{b}-y=-a$
      $\frac{(b^2-a^2)x}{ab}=1-a$
      $x=\frac{ab-a^{2}b}{b^2-a^2}$
   2. Substitute solution for x into the first equation to solve for y.
      $\frac{ab-a^{2}b}{a(b^2-a^2)}+\frac{y}{b}=1$
      $\frac{b-ab}{b^2-a^2}+\frac{y}{b}=1$
      $y=\frac{a{b}^2-b^2}{b^2-a^2}$
4. Solve each of the following pairs of equations for $x$ and $y$

   1. $y=ax-b$ and $y=cx+a$
      $ax-b=cx+a$
      $x(a-c)=a+b$
      $x=\frac{a+b}{a-c}$
      $y=a(\frac{a+b}{a-c})-b$
      $y=\frac{a^2+ba-b(a-c)}{a-c}$
      $y=\frac{a^2+bc}{a-c}$

   2. $ax+y=c$ and $x+by=d$
      $x+b(c-ax)=d$
      $x+bc-abx=d$
      $x(1-ab)=d-bc$
      $x=\frac{d-bc}{1-ab}$
      $ax+y=c$
      $ax+aby=ad$
      $y-aby=c-ad$
      $y=\frac{c-ad}{1-ab}$

   3. $ax-by=a^2$ and $bx-ay=b^2$
      $a^{2}x-aby=a^3$
      $b^{2}x-aby=b^3$
      $a^{2}x-b^{2}x=a^3-b^3$
      $x=\frac{a^3-b^3}{a^2-b^2}$
      $x=\frac{(a-b)(a^2+ab+b^2)}{(a+b)(a-b)}$
      $x=\frac{a^2+ab+b^2}{a+b}$
      $abx-b^{2}y=a^{2}b$
      $abx-a^{2}y=a{b}^2$
      $-b^{2}y+a^{2}y=a^{2}b-a{b}^2$
      $y=\frac{a^{2}b-a{b}^2}{a^2-b^2}$
      $y=\frac{(ab)(a-b)}{(a+b)(a-b)}$
      $y=\frac{ab}{a+b}$

   4. $ax+by=t$ and $ax-by=s$
      $2by=t-s$
      $y=\frac{t-s}{2b}$
      $2ax=t+s$
      $x=\frac{t+s}{2a}$

   5. $y=abx+ac$ and $y=ax-bc$
      $by=abx-b^{2}c$
      $y-by=ac+b^{2}c$
      $y=\frac{ac+b^{2}c}{1-b}$
      $abx+ac=ax-bc$
      $x(ab-a)=-bc-ac$
      $x=\frac{ac+bc}{a-ab}$

   6. $ax+by=a^2+2ab-b^2$ and $bx+ay=a^2+b^2$
      $abx+b^{2}y=b(a^2+2ab-b^2)$
      $abx+a^{2}y=a(a^2+b^2)$
      $b^{2}y-a^{2}y=b{a}^2+2a{b}^2-b^3-(a^3+a{b}^2)$
      $y(b^2-a^2)=a^{2}b+2a{b}^2-b^3-a^3-a{b}^2$
      $y(b^2-a^2)=a^{2}b+a{b}^2-a^3-b^3$
      $y=\frac{a^{2}b+a{b}^2-a^3-b^3}{b^2-a^2}$
      $y=\frac{(ab)(a+b)-a^3-b^3}{(b+a)(b-a)}$
      $y=\frac{(ab)(a+b)+(-1)(a+b)(a^2-ab+b^2)}{(b+a)(b-a)}$
      $y=\frac{ab-a^2+ab-b^2}{(b-a)}$
      $y=\frac{(-1)(a^2-2ab+b^2)}{(b-a)}$
      $y=\frac{(-1)(a-b)(a-b)}{(b-a)}$
      $y=a-b$

      $bx+a(a-b)=a^2+b^2$
      $bx+a^2-ab=a^2+b^2$
      $bx=b^2+ab$
      $x=a+b$

5. For each of the following, eliminate $h$ and express $s$ in terms of $a$
   1. $s=ah$ and $h=2a+1$
      $h=\frac{s}{a}$
      $\frac{s}{a}=2a+1$
      $s=2a^2+a$
   2. $s=ah$ and $h=a(2+h)$
      $h=\frac{s}{a}$
      $\frac{s}{a}=2a+\frac{as}{a}$
      $\frac{s}{a}=2a+s$
      $\frac{s-sa}{a}=2a$
      $s(1-a)=2a^2$
      $s=\frac{2a^2}{1-a}$
   3. $s=h^2+ah$ and $h=3a^2$
      $s=9a^4+3a^3$
   4. $as=a+h$ and $h+ah=1$
      $h=\frac{1}{1+a}$
      $as=a+\frac{1}{1+a}$
      $s=1+\frac{1}{x^2+a}$
      $s=\frac{a^2+a+1}{a^2+a}$
   5. $as=s+h$ and $ah=a+h$
      $h=\frac{a}{a-1}$
      $as=s+\frac{a}{a-1}$
      $as-s=\frac{a}{a-1}$
      $s(a-1)=\frac{a}{a-1}$
      $s=\frac{a}{(a-1{)}^2}$
   6. $as=a+2h$ and $h=a-s$
      $h=\frac{as-a}{2}$
      $\frac{as-a}{2}=a-s$
      $as-a=2a-2s$
      $as+2s=3a$
      $s=\frac{3a}{a+2}$