Algebra - Algorithms

Solve the linear equation for $x$: $4(x - 3) = 2x + 10$

$4x - 12 = 2x + 10$ $4x - 2x = 10 + 12$ $2x = 22$ $x = 11$

Apply the expansion algorithm to the left side: $4 \times x$ and $4 \times -3$. Then, use the balancing algorithm to group like terms by subtracting $2x$ and adding $12$ to both sides. Finally, divide by the coefficient of $x$.

Factorize the quadratic expression: $x^2 - 7x + 12$

$(x - 3)(x - 4)$ patterns.

Use the 'Sum and Product' algorithm. We look for two numbers that multiply to $+12$ (the constant) and add to $-7$ (the coefficient of $x$). Those numbers are $-3$ and $-4$. Thus, the factors are $(x - 3)$ and $(x - 4)$.

Solve the simultaneous equations using the Elimination Algorithm: $x + y = 15$ $x - y = 3$

$\begin{array}{r} x + y = 15 \\ + (x - y = 3) \\ \hline 2x = 18 \end{array}$ $x = 9$ $9 + y = 15 \Rightarrow y = 6$

The algorithm involves adding the two equations to eliminate the variable $y$. This results in a single-variable equation for $x$. Once $x$ is found, substitute it back into one of the original equations to solve for $y$.

Make $h$ the subject of the formula: $V = \frac{1}{3} π r^2 h$

$3V = π r^2 h$ $h = \frac{3V}{π r^2}$

To isolate $h$, apply the inverse operation algorithm. First, multiply both sides by $3$ to remove the fraction. Then, divide both sides by the product $π r^2$ to leave $h$ alone.