Evaluate $\displaystyle \quad \lim_{x\to a} \frac{ax^2+bx+c}{ex+d} \quad $ algebraically.

Evaluate $\displaystyle \quad \lim_{x\to a} \frac{ax+b}{x^2+c} \quad $ algebraically.

Students will solve for x using algebraic maipulation.

This question uses the linear algebra extension to generate a system of linear equations which can be solved.

We want to produce an equation of the form $\mathrm{A}\mathbf{x} = \mathbf{y}$, where $\mathrm{A}$ and $\mathbf{y}$ are given, and $\mathbf{x}$ is to be found by the student.

First, we generate a linearly independent set of vectors to form $\mathrm{A}$, then freely pick the value of $\mathbf{x}$, and calculate the corresponding $\mathbf{y}$.

To generate $\mathrm{A}$, we generate more vectors we need, then pick a linearly independent subset of those using the subset_with_dimension function.

Simplifying algebraic expressions

Arithmettik & Algebra in der Renaissance

Arithmetik & Algebra in der Renaissance

Adaptiert von: Katz, V. J. (2009). A history of mathematics: An introduction (3. ed.). Boston: Addison-Wesley, S. 418.

Simplificación de expresiones algebraicas

Add, subtract, multiply and divide algebraic fractions.

Add, subtract, multiply and divide algebraic fractions.

Inputting algebraic expressions into Numbas. (Translation to German)

Students seem to not realise that $\frac{a}{b}\times c=c\times\frac{a}{b}=\frac{a\times c}{b}=\frac{c\times a}{b}=a\times c \div b=a\div b\times c=c\div b \times a \ne c \div (b\times a)\ldots $ etc. This question is my attempt to help rectify this.

This question tests the student's ability to solve simple linear equations by elimination. Part a) involves only having to manipulate one equation in order to solve, and part b) involves having to manipulate both equations in order to solve. 

Solve for $x$: $\displaystyle \frac{px+s}{ax+b} = \frac{qx+t}{cx+d}$ with $pc=qa$.

German translation of https://numbas.mathcentre.ac.uk/question/12012/solve-an-equation-in-algebraic-fractions/ by Newcastle University Mathematics and Statistics

Algebra:  Quadratic Factorisation.

              Coefficient of squared term is 1.

Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Students are asked to obtain a percentage from an infographic, then choose the fraction which is closest. There are 7 different versions of this question.

Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Students are given the gifted proportion as a decimal, and the proportion of those who receive support as a decimal, and are required to find the proportion of the whole school who are gifted but do not receive support (as a decimal). There are 9 versions of this question.

Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Students are given two probabilities in ratio form, and are required to select the higher probability. There are 7 different versions of this question.

This question was modified from a Newcastle University question. Used for LANTITE preparation (Australia). NC = Non Calculator strand. NA = Number & Algebra strand. Students need to work out a ratio and simplify it.

Three lines given with different information provided. Equation of the line is asked for.

A question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).

A set of Numbas exercises for students transitioning from school to University. Designed to help students gain familiarity with using Numbas to enter mathematics, and as revision for algebra, geometry and calculus. 

Practise solving simultaneous linear equations graphically and algebraically.

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.

This quiz contains questions on algebraic fractions, logarithmic equations, exponential equations, quadratic equations and simultaneous equations.

Splitting an algebraic fraction into partial fractions

rebelmaths

Express $\log_a(x^{c}y^{d})$ in terms of $\log_a(x)$ and $\log_a(y)$. Find $q(x)$ such that $\frac{f}{g}\log_a(x)+\log_a(rx+s)-\log_a(x^{1/t})=\log_a(q(x))$.

There is a video included explaining the rules of logarithms by going through simplification of logs of numbers rather than algebraic expressions.

Algebraic manipulation/simplification.

Simplify $\displaystyle \frac{ax^4+bx^2+c}{a_1x^4+b_1x^2+c_1}$ by cancelling a a common degree 2 factor.

Simplify $(ax+b)(cx+d)-(ax+d)(cx+b)$. Answer is a multiple of $x$.