Solve the following pair of linear equations by the elimination method and the substitution method:
(i) \(x + y = 5\) and \(2x - 3y = 4\)
(ii) \(3x + 4y = 10\) and \(2x - 2y = 2\)
(iii) \(3x - 5y - 4 = 0\) and \(9x = 2y + 7\)
(iv) \(\dfrac{x}{2} + \dfrac{2y}{3} = -1\) and \(x - \dfrac{y}{3} = 3\)
(i) \(x + y = 5\), \(2x - 3y = 4\)
From \(x + y = 5\), express \(x\) as \(x = 5 - y\). Substitute in \(2x - 3y = 4\):
\[2(5 - y) - 3y = 4 \Rightarrow 10 - 2y - 3y = 4 \Rightarrow 10 - 5y = 4.\]
\[-5y = -6 \Rightarrow y = \dfrac{6}{5}.\]
Then \(x = 5 - y = 5 - \dfrac{6}{5} = \dfrac{25 - 6}{5} = \dfrac{19}{5}.\)
Solution: \(x = \dfrac{19}{5},\ y = \dfrac{6}{5}.\)
(ii) \(3x + 4y = 10\), \(2x - 2y = 2\)
From \(2x - 2y = 2\), divide by 2:
\[x - y = 1 \Rightarrow x = y + 1.\]
Substitute in \(3x + 4y = 10\):
\[3(y + 1) + 4y = 10 \Rightarrow 3y + 3 + 4y = 10 \Rightarrow 7y + 3 = 10.\]
\[7y = 7 \Rightarrow y = 1, \quad x = y + 1 = 2.\]
Solution: \(x = 2,\ y = 1.\)
(iii) \(3x - 5y - 4 = 0\) and \(9x = 2y + 7\)
Rewrite the equations:
\[3x - 5y = 4, \quad 9x - 2y = 7.\]
Multiply the first equation by 2:
\[6x - 10y = 8.\]
Multiply the second equation by 5:
\[45x - 10y = 35.\]
Subtract the first (multiplied) equation from the second:
\[(45x - 10y) - (6x - 10y) = 35 - 8 \Rightarrow 39x = 27.\]
\[x = \dfrac{27}{39} = \dfrac{9}{13}.\]
Substitute in \(3x - 5y = 4\):
\[3 \cdot \dfrac{9}{13} - 5y = 4 \Rightarrow \dfrac{27}{13} - 5y = 4.\]
\[-5y = 4 - \dfrac{27}{13} = \dfrac{52 - 27}{13} = \dfrac{25}{13} \Rightarrow y = -\dfrac{25}{65} = -\dfrac{5}{13}.\]
Solution: \(x = \dfrac{9}{13},\ y = -\dfrac{5}{13}.\)
(iv) \(\dfrac{x}{2} + \dfrac{2y}{3} = -1\), \(x - \dfrac{y}{3} = 3\)
Clear denominators by multiplying each equation by 6:
First equation:
\[6\left(\dfrac{x}{2} + \dfrac{2y}{3}\right) = 6(-1) \Rightarrow 3x + 4y = -6.\]
Second equation:
\[6\left(x - \dfrac{y}{3}\right) = 6 \cdot 3 \Rightarrow 6x - 2y = 18.\]
Divide the second equation by 2 to simplify:
\[3x - y = 9.\]
Now solve the pair \(3x + 4y = -6\) and \(3x - y = 9\) by elimination. Subtract the second from the first:
\[(3x + 4y) - (3x - y) = -6 - 9 \Rightarrow 5y = -15 \Rightarrow y = -3.\]
Substitute in \(3x - y = 9\):
\[3x - (-3) = 9 \Rightarrow 3x + 3 = 9 \Rightarrow 3x = 6 \Rightarrow x = 2.\]
Solution: \(x = 2,\ y = -3.\)
Clear decimals or fractions, then use substitution or elimination to remove one variable. Solve the simplified equation, back-substitute, and confirm the ordered pair satisfies both equations.