The condition for which the pair of equations ax+2y=7 and 3x+by=16 represent parallel lines is :
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Step-by-Step Solution
Step 1: Rewrite equations in standard form
First, we rewrite the given linear equations in the standard form Ax+By+C=0. This helps in easily identifying the coefficients of x and y, and the constant terms.
Step 2: Identify coefficients
From the rewritten equations, we can identify the coefficients for each variable and the constant terms. For the first equation, a1=a, b1=2, and c1=−7. For the second equation, a2=3, b2=b, and c2=−16.
Step 3: Apply condition for parallel lines
For two linear equations to represent parallel lines, the ratio of their x coefficients must be equal to the ratio of their y coefficients, but not equal to the ratio of their constant terms. This condition ensures that the lines have the same slope but different y -intercepts.
Step 4: Substitute coefficients and solve
Substitute the identified coefficients into the condition for parallel lines. We get 3a=b2 and 3a=167. From the first equality, we cross-multiply to find the relationship between a and b.
Step 5: Derive the condition
From the equality 3a=b2, cross-multiplying gives a×b=3×2, which simplifies to ab=6. This is the required condition for the lines to be parallel.