Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>If the system of equations <span class="math">ax + y = 1</span>, <span class="math">x + 2y = 3</span>, <span class="math">2x + 3y = 5</span> are consistent, then <span class="math">a</span> is given by</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: For a system of equations to be consistent, all three equations must be satisfied by the same values of variables. Solve any two equations first, then check the third.
<p>For the system to be consistent, the equations must have a solution. From the second and third equations, we can solve for <span class="math">x</span> and <span class="math">y</span>. From <span class="math">x + 2y = 3</span> and <span class="math">2x + 3y = 5</span>, we get <span class="math">x = 1, y = 1</span>. Substituting in the first equation: <span class="math">a(1) + 1 = 1</span>, so <span class="math">a = 0</span>.</p>
Correct Answer: a