Lesson 3 · from §86 of the 1910 text
Systems Containing Fractions
Systems containing fractions
Example. Solve \begin{cases} \dfrac{8x}{3} - \dfrac{59}{6} = \dfrac{3y}{2} & (1) \\[2mm] \dfrac{3x}{4} = -2y - \dfrac92 & (2) \end{cases}
| Step | Result | |
|---|---|---|
| (1)\cdot 6 | 16x - 59 = 9y | (3) |
| transposing | 16x - 9y = 59 | (4) |
| (2)\cdot 4 | 3x = -8y - 18 | (5) |
| transposing | 3x + 8y = -18 | (6) |
| (4)\cdot 3 | 48x - 27y = 177 | (7) |
| (6)\cdot 16 | 48x + 128y = -288 | (8) |
| (7) - (8) | -155y = 465 | (9) |
So y = -3, and from (4), 16x + 27 = 59, so x = 2.
Check in (1): \frac{16}{3} - \frac{59}{6} = -\frac92 and \frac{3(-3)}{2} = -\frac92. ✓ In (2): \frac64 = 6 - \frac92 = \frac32. ✓
As here, it is usually best to clear each equation of fractions and write it in the form of (4) and (6) before eliminating. That form, ax + by = c, is the general form of a linear equation in two variables; a, b, c stand for numbers or known literal expressions.
Letters in the denominator
When the unknowns are in the denominators, as in \frac1x + \frac1y = \frac56 and \frac1x - \frac1y = \frac16, don't clear of fractions: treat \frac1x and \frac1y as the unknowns. Adding, \frac2x = 1, so x = 2; then \frac1y = \frac13 and y = 3.
A value that makes a denominator zero is never a root. If the only values you find do that, the system has no set of roots: type none.
How to type your answers
Type each value in its own box: a whole number, a fraction like -17/5, or a mixed number like 4 1/2.
- If the system is inconsistent (no set of roots), type
nonein every box. - If it has an unlimited number of sets of roots, type
anyin the box for the first letter and give the other letter(s) in terms of it.
Try one:
Quick check
Solve \begin{cases} \dfrac x2 + \dfrac y3 = 4 \\[2.5mm] x - y = 3 \end{cases}
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Solve each system and check.
\begin{cases} \dfrac{2x}{3} + 4y = \dfrac{26}{3} \\[2.5mm] 3x - \dfrac{7y}{2} = -4 \end{cases}
\begin{cases} m - \dfrac{3n}{5} = \dfrac{18}{5} \\[2.5mm] \dfrac{8n}{3} + 7m = -16 \end{cases}
\begin{cases} \dfrac{5R_1}{6} + \dfrac{R_2}{4} = 7 \\[2.5mm] \dfrac{2R_1}{3} - \dfrac{R_2}{8} = 3 \end{cases}
\begin{cases} 3k - \dfrac{12h}{5} = 18 \\[2.5mm] \dfrac{11k}{10} + \dfrac{17h}{2} = 53\tfrac{1}{2} \end{cases}
\begin{cases} \dfrac{2x}{9} - \dfrac{y}{2} = -1 \\[2.5mm] x = \dfrac{9y}{4} - 4\tfrac{1}{2} \end{cases}
\begin{cases} .4x + .9y = 5.7 \\[2.5mm] 2x - y = 1 \end{cases}
\begin{cases} 12m = \dfrac{11y}{9} + 17 \\[2.5mm] \dfrac{2m}{3} - \dfrac{49}{12} = \dfrac{5y}{12} \end{cases}
\begin{cases} 9s = \dfrac{t}{3} + \dfrac{23}{4} \\[2.5mm] \dfrac{7t}{3} - \dfrac{9}{4} = \dfrac{-3s}{4} \end{cases}
\begin{cases} .04m + .75n = 10 \\[2.5mm] .8m - 1.25n = 5 \end{cases}
\begin{cases} 28x - 16y = 56 \\[2.5mm] \dfrac{2x + 7}{2} - y = 4 \end{cases}
\begin{cases} \dfrac{r + 3}{7} - \dfrac{l}{5} = 0 \\[2.5mm] 5r + 2 = 7l \end{cases}
\begin{cases} \dfrac{x + 3}{5} + \dfrac{y + 4}{10} = \dfrac32 \\[2.5mm] \dfrac{7x + 1}{3} - \dfrac{11y - 4}{7} = 4 \end{cases}
\begin{cases} \dfrac{20m + 9}{7} = \dfrac{n}{7} + 1 \\[2.5mm] m + \dfrac{29}{5} = n \end{cases}
\begin{cases} \dfrac1x + \dfrac1y = \dfrac56 \\[2.5mm] \dfrac1x - \dfrac1y = \dfrac16 \end{cases}
\begin{cases} \dfrac{1}{m_1} + \dfrac{1}{m_2} = \dfrac{2}{15} \\[2.5mm] \dfrac{1}{m_1} - \dfrac{1}{m_2} = -\dfrac13 \end{cases}
\begin{cases} \dfrac6x + \dfrac7y = \dfrac32 \\[2.5mm] \dfrac7x - \dfrac6y + 5\tfrac{1}{3} = 0 \end{cases}
\begin{cases} \dfrac1x + \dfrac1y = 10 \\[2.5mm] \dfrac3x + \dfrac3y = 14 \end{cases}
\begin{cases} \dfrac5x + 12 = 17 \\[2.5mm] \dfrac2x - 15y = 0 \end{cases}
\begin{cases} \dfrac{4r + q}{6r + q} = \dfrac25 \\[2.5mm] \dfrac{2r}{3} - q = 5 \end{cases}
\begin{cases} \dfrac{1}{t_2} + \dfrac{2}{t_1} = \dfrac{40}{t_1 t_2} \\[2.5mm] 2t_2 - 3t_1 = 0 \end{cases}
\begin{cases} \dfrac{5m + 3n - 1}{3} = n + \dfrac13 \\[2.5mm] \dfrac{25m}{10m + 2n} = 1 \end{cases}
\begin{cases} \dfrac{k + 5}{k + 1} = \dfrac{l + 2}{l - 2} \\[2.5mm] \dfrac{11k + 4}{l + k + 1} = 4 \end{cases}
\begin{cases} \dfrac{2x_1 - 3x_2}{x_2 - 4x_1} = 3 \\[2.5mm] \dfrac{8}{x_2 + x_1} + \dfrac{8}{x_1 - x_2} = 0 \end{cases}
\begin{cases} 3x - 2y = 7 \\[2.5mm] \dfrac{2}{x - 5} = -\dfrac{5}{y - 4} \end{cases}
\begin{cases} \dfrac{2}{6 + s} - \dfrac{4}{4 - t} = 0 \\[2.5mm] \dfrac t4 + \dfrac s2 + 2 = 0 \end{cases}
\begin{cases} \dfrac{2}{n_1 + 2 + n_2} = \dfrac{1}{5n_2 + 3n_1 - 5} \\[2.5mm] n_1 + 2n_2 = 4 \end{cases}