The number of solutions to the pair of linear equations 5x – 3y = 7 and 7x + 4y = 18 is:

The pair of linear equations:

5x − 3y = 7

7x + 4y = 18

To determine the number of solutions for a pair of linear equations:

The general form of two linear equations is:

​$$a_1x + b_1y = c_1 \quad \text{and} \quad a_2x + b_2y = c_2$$​​

Conditions for the solutions:

Unique solution:

​$$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$​​

Infinitely many solutions:

If  $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$​ .

No solution:

If $$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$​​

From the given equations:

​$$a_1 = 5a, \ \ \ b_1 = -3, \ \ \ c_1 = 7$$​​

​$$a_2 = 7,\ \ \ b_2 = 4,\ \ \ c_2 = 18$$​​

Now:

​$$\frac{a_1}{a_2} = \frac{5}{7}$$​​

​$$\frac{b_1}{b_2} = \frac{-3}{4}$$​​

Since:

​$$\frac{5}{7} \neq \frac{-3}{4},​$$​​

the lines are not parallel, and they intersect at exactly one point.

Thus, the lines have only one unique solution.