Question:
<p>If the system of equations<br />
2x + 3y = 7<br />
2ax + (a + b)y = 28<br />
has infinitely many solutions, then the values of a and b respectively are ________.</p>
<p style="display:inline">4, 8</p>
<p style="display:inline">2, 5</p>
<p style="display:inline">5, 8</p>
<p style="display:inline">3, 6</p>
Step-by-Step Solution
Key Concept: For a system of linear equations to have infinitely many solutions, the ratios of the coefficients of x, the coefficients of y, and the constant terms must all be equal.
<p>Given equations are<br />
2x + 3y = 7 and 2ax + (a + b)y = 28<br />
For infinitely many solutions, we have<br />
<span class="math-tex">$\frac{2}{2 a}$</span> = <span class="math-tex">$\frac{3}{a+b}$</span> = <span class="math-tex">$\frac{7}{28}$</span><br />
<span class="math-tex">$\Rightarrow$</span> 2a + 2b = 6a<br />
<span class="math-tex">$\Rightarrow$</span> 4a = 2b <span class="math-tex">$\Rightarrow$</span> 2a = b ...(i)<br />
Also, <span class="math-tex">$\frac{2}{2a}$</span> = <span class="math-tex">$\frac{7}{28}$</span> <span class="math-tex">$\Rightarrow$</span> a = 4 ...(ii)<br />
From (i) and (ii), we have<br />
2(4) = b <span class="math-tex">$\Rightarrow$</span> b = 8</p>
Correct Answer: A