Lesson 2 · from §85 of the 1910 text
Solving by Substitution
Solution by substitution
Example. Solve \begin{cases} 3x - 13y = 41 & (1) \\ 8x + 11y = 18 & (2) \end{cases}
From (1), 3x = 13y + 41, so x = \dfrac{13y + 41}{3}. (3)
Substituting this for x in (2): \;8\cdot\dfrac{13y + 41}{3} + 11y = 18.
Multiplying by 3: 8(13y + 41) + 33y = 54, so 104y + 328 + 33y = 54, 137y = -274, and y = -2.
Substituting -2 for y in (3): x = \dfrac{-26 + 41}{3} = 5.
Check: 15 + 26 = 41 and 40 - 22 = 18. ✓
Rule. Solve either equation for one variable in terms of the other. Substitute this value in the equation from which it was not obtained and solve. Substitute the value just found in the simplest equation containing both variables and solve for the other.
Check as before.
Substitution is handiest when one equation already gives one letter in terms of the other, like t = s + 1 or q = p - 16.
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 by substitution: \begin{cases} y = 2x - 1 \\[2.5mm] 3x + y = 9 \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 by the method of substitution.
\begin{cases} x - 2y = 8 \\[2.5mm] 3x + 2y = 7 \end{cases}
\begin{cases} x - 2y = -12 \\[2.5mm] 4x - y = 1 \end{cases}
\begin{cases} 14m - 2n = 1 \\[2.5mm] n - 6m = 0 \end{cases}
\begin{cases} 6h + 10k = 19 \\[2.5mm] 2k = 3h \end{cases}
\begin{cases} 3s + 12 = 3 + t \\[2.5mm] t = s + 1 \end{cases}
\begin{cases} 18 + 2p = q \\[2.5mm] p + q = -9 \end{cases}
\begin{cases} 3r + 15s = 7 \\[2.5mm] 12 + 5s = -r \end{cases}
\begin{cases} 20y - 3z = 1 \\[2.5mm] z - 6y = 0 \end{cases}
\begin{cases} .75p + 1.5q = 3 \\[2.5mm] q = p - 16 \end{cases}
\begin{cases} \dfrac{3x - 20}{2} = \dfrac{2x + 5y}{3} \\[2.5mm] 10 = x - y \end{cases}
\begin{cases} \dfrac{1}{7 + 2m_1} = \dfrac{-7}{m_2 - 1} \\[2.5mm] m_1 = m_2 + 3 \end{cases}
\begin{cases} \dfrac{5R_1 + 2R_2}{2} = 2(R_2 + 2) \\[2.5mm] R_1 - \dfrac23 R_2 = 0 \end{cases}