Question

Solve the following pairs of linear equations : $3x - 5y = 4$, $9x = 2y + 7$

Hint:

Substitution would also be efficient here by substituting $9x$ from Eq. 2 into a tripled version of Eq. 1.

Answer 1

Step 1: Write the given equations.
The pair of linear equations is:
\(3x - 5y = 4\) …… (1)
\(9x = 2y + 7\) …… (2)

First rewrite equation (2) in standard form:
\(9x - 2y = 7\) …… (2)

Step 2: Use the elimination method.
We eliminate one variable. Multiply equation (1) by 3 so that the coefficient of \(x\) becomes equal in both equations.

Multiply (1) by 3:
\(9x - 15y = 12\) …… (3)

Now subtract equation (2) from equation (3).

\((9x - 15y) - (9x - 2y) = 12 - 7\)

Step 3: Simplify the equation.
\(9x - 15y - 9x + 2y = 5\)
\(-13y = 5\)
\(y = -\frac{5}{13}\)

Step 4: Substitute the value of \(y\) into one equation.
Substitute \(y = -\frac{5}{13}\) into equation (1):
\(3x - 5\left(-\frac{5}{13}\right) = 4\)
\(3x + \frac{25}{13} = 4\)

Convert 4 into fraction form:
\(4 = \frac{52}{13}\)

\(3x + \frac{25}{13} = \frac{52}{13}\)

Step 5: Solve for \(x\).
\(3x = \frac{52}{13} - \frac{25}{13}\)
\(3x = \frac{27}{13}\)
\(x = \frac{27}{39}\)
\(x = \frac{9}{13}\)

Final Answer:
\(x = \frac{9}{13}\)
\(y = -\frac{5}{13}\).