Semi-worked example of solving simultaneous equations using matrices. Equation values are randomly generated. The student is walked through the steps needed to solve the equations.

Solve unknown on one side

Solve $\displaystyle ay + b = cy + d$ for $y$.

Solve an exponential equation

Students need to substitute a value into an equation and solve it. The equation constant (gravity) and the value (time) are both randomised.

Students are given 2 equations of the form y=mx+b and asked to solve them using either the substitution or the elimination method. The lines are randomised but the solution coordinates are always integers.

Solve for $x$:  $\log_{a}(x+b)- \log_{a}(x+c)=d$

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Solve $\displaystyle ax + b = cx + d$ for $x$.

Solve for $x$: $\displaystyle \frac{a} {bx+c} + d= s$

Solve $\displaystyle ax + b =\frac{f}{g}( cx + d)$ for $x$.

A video is included in Show steps which goes through a similar example.

Differentiate $f(x)=x^{m}\sin(ax+b) e^{nx}$.

The answer is of the form:
$\displaystyle \frac{df}{dx}= x^{m-1}e^{nx}g(x)$ for a function $g(x)$.

Find $g(x)$.

Solve for $x$:  $\log_{a}(x+b)- \log_{a}(x+c)=d$

Solve for $x$: $\displaystyle 2\log_{a}(x+b)- \log_{a}(x+c)=d$. 

Make sure that your choice is a solution by substituting back into the equation.

Solve for $x$: $\log(ax+b)-\log(cx+d)=s$

Solve for $x$ each of the following equations: $n^{ax+b}=m^{cx}$ and $p^{rx^2}=q^{sx}$.

Solve for $x$: $c(a^2)^x + d(a)^{x+1}=b$ (there is only one solution for this example).

Solve for $x$: $a\cosh(x)+b\sinh(x)=c$. There are two solutions for this example.

Solve for $x$: $\displaystyle ax ^ 2 + bx + c=0$.

Differentiate $x^m\cos(ax+b)$

Differentiate $ (a+bx) ^ {m} \sin(nx)$

Differentiate $ (ax+b)^m(cx+d)^n$ using the product rule. The answer will be of the form $(ax+b)^{m-1}(cx+d)^{n-1}g(x)$ for a polynomial $g(x)$. Find $g(x)$.

Differentiate $ x ^ m(ax+b)^n$ using the product rule. The answer will be of the form $x^{m-1}(ax+b)^{n-1}g(x)$ for a polynomial $g(x)$. Find $g(x)$.

Differentiate $f(x) = x^m(a x+b)^n$.

Differentiate $ x ^m \sqrt{a x+b}$.
The answer is in the form $\displaystyle \frac{x^{m-1}g(x)}{2\sqrt{ax+b}}$
for a polynomial $g(x)$. Find $g(x)$.

Differentiate the function $(a + b x)^m  e ^ {n x}$ using the product rule.

Inverse and division of complex numbers.  Four parts.

Multiplication of complex numbers. Four parts.

Find $p$ and $q$ such that $ax^2+bx+c = a(x+p)^2+q$. 

Hence, or otherwise, find roots of  $ax^2+bx+c=0$.

Includes a video which shows how to solve a quadratic by completing the square.

4 questions on using partial fractions to solve indefinite integrals.