Equations Two Unknowns

Overview

Engineering reference data for Equations Two Unknowns in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Solution by Substitution

For a system of first-degree equations with two unknowns: $a_1 x + b_1 y = c_1 \quad (1)$ $a_2 x + b_2 y = c_2 \quad (2)$

The solution can be found by substitution:

1. Solve equation (1) for $y$: $y = \frac{c_1 - a_1 x}{b_1}$
2. Substitute into equation (2): $a_2 x + b_2 \left( \frac{c_1 - a_1 x}{b_1} \right) = c_2$
3. Solve for $x$: $x = \frac{c_2 b_1 - c_1 b_2}{a_2 b_1 - a_1 b_2}$
4. Back-substitute to find $y$: $y = \frac{c_1 a_2 - c_2 a_1}{a_2 b_1 - a_1 b_2}$

Example Calculation

Consider the system: $3x + 5y = 7$ $4x - 3y = 19$

Using substitution:

1. From the first equation: $y = \frac{7 - 3x}{5}$
2. Substitute into the second equation: $4x - 3\left(\frac{7 - 3x}{5}\right) = 19$
3. Solve for $x$: $4x - \frac{21 - 9x}{5} = 19$ Multiply by 5: $20x - (21 - 9x) = 95$ $20x - 21 + 9x = 95$ $29x = 116$ $x = 4$
4. Substitute $x = 4$ back: $3(4) + 5y = 7$ $12 + 5y = 7$ $5y = -5$ $y = -1$

Thus, the solution is $x = 4$, $y = -1$.

References

• Original Source