FORMULA CHAPTER
\begin{aligned}a_1x+b_1y&=c_1\\a_2x+b_2y&=c_2\end{aligned}
\frac{a_1}{a_2}\ne\frac{b_1}{b_2}
\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}
\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}
x=\frac{c_1b_2-c_2b_1}{a_1b_2-a_2b_1}
y=\frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}
\begin{aligned}a_1x+b_1y&=c_1\\a_2x+b_2y&=c_2\end{aligned}
Question: Express 2x + 3y = 7 and 4x − y = 5 in standard simultaneous-equation form and identify the coefficients.
Calculation:
Step 1: Compare 2x + 3y = 7 with a₁x + b₁y = c₁.
a₁ = 2, b₁ = 3, c₁ = 7.
Step 2: Compare 4x − y = 5 with a₂x + b₂y = c₂.
a₂ = 4, b₂ = −1, c₂ = 5.
Answer: The equations are in standard form with coefficients (2, 3, 7) and (4, −1, 5).
\frac{a_1}{a_2}\ne\frac{b_1}{b_2}
Question: Determine the nature of the solution of 2x + 3y = 7 and 4x − y = 5.
Calculation:
Step 1: a₁/a₂ = 2/4 = 1/2.
Step 2: b₁/b₂ = 3/(−1) = −3.
Step 3: Since 1/2 ≠ −3, a₁/a₂ ≠ b₁/b₂.
Answer: The equations have one unique solution.
\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}
Question: Determine the nature of the solution of x + y = 2 and 2x + 2y = 4.
Calculation:
Step 1: a₁/a₂ = 1/2.
Step 2: b₁/b₂ = 1/2.
Step 3: c₁/c₂ = 2/4 = 1/2.
Step 4: All three ratios are equal.
Answer: The equations represent the same line and have infinitely many solutions.
\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}
Question: Determine the nature of the solution of x + y = 2 and 2x + 2y = 5.
Calculation:
Step 1: a₁/a₂ = 1/2 and b₁/b₂ = 1/2.
Step 2: c₁/c₂ = 2/5.
Step 3: Therefore a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
Answer: The equations represent distinct parallel lines and have no solution.
x=\frac{c_1b_2-c_2b_1}{a_1b_2-a_2b_1}
Question: Find x for 2x + y = 7 and x − y = 2 using cross multiplication.
Calculation:
Step 1: a₁ = 2, b₁ = 1, c₁ = 7; a₂ = 1, b₂ = −1, c₂ = 2.
Step 2: Numerator = c₁b₂ − c₂b₁ = 7(−1) − 2(1) = −9.
Step 3: Denominator = a₁b₂ − a₂b₁ = 2(−1) − 1(1) = −3.
Step 4: x = (−9)/(−3) = 3.
Answer: x = 3.
y=\frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}
Question: Find y for 2x + y = 7 and x − y = 2 using cross multiplication.
Calculation:
Step 1: a₁ = 2, b₁ = 1, c₁ = 7; a₂ = 1, b₂ = −1, c₂ = 2.
Step 2: Numerator = a₁c₂ − a₂c₁ = 2(2) − 1(7) = −3.
Step 3: Denominator = a₁b₂ − a₂b₁ = 2(−1) − 1(1) = −3.
Step 4: y = (−3)/(−3) = 1.
Answer: y = 1.
