Algebra

The One Thing

Algebra represents mathematical relationships symbolically and solves them through valid, consistent operations. Its main applications include equations and inequalities, ratios and proportions, factorisation, matrices, progressions, and coordinate geometry.

Definitions and Results

• Linear inequation: An inequality involving a variable of degree one, using symbols such as $<$, $>$, $\le$, or $\ge$.
• Solution set: The set of all values of the variable that make an equation or inequation true.
• Quadratic equation: An equation of the form $a{x}^2+bx+c=0$, where $a$, $b$, and $c$ are real numbers and $a\ne 0$.
• Discriminant: The expression $b^2-4ac$ in a quadratic equation. It indicates the nature of the roots.
• Ratio: A comparison of two quantities measured in the same units, written as $a:b$.
• Proportion: A statement that two ratios are equal, such as $a:b=c:d$.
• Factorisation: Writing an algebraic expression as a product of simpler factors.
• Matrix: A rectangular arrangement of numbers or algebraic expressions in rows and columns.
• Order of a matrix: The number of rows followed by the number of columns, written as $m\times n$.
• Arithmetic progression: A sequence in which the difference between consecutive terms is constant.
• Geometric progression: A sequence in which the ratio between consecutive non-zero terms is constant.
• Coordinate geometry: The study of geometric figures using points, coordinates, equations, and algebraic formulas.
• Distance formula: A formula used to find the distance between two points on the Cartesian plane.
• Section formula: A formula used to find the coordinates of a point dividing a line segment internally or externally in a given ratio.
• Slope: The rate of change of a line, found from the change in $y$-coordinates divided by the change in $x$-coordinates.

• Operations on linear inequations: Adding or subtracting the same number on both sides preserves the inequality. Multiplying or dividing both sides by a negative number reverses the inequality sign.
• Quadratic formula: For $a{x}^2+bx+c=0$,

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

• Discriminant conditions: If $D=b^2-4ac$, then:

- $D>0$: two distinct real roots; - $D=0$: two equal real roots; - $D<0$: no real roots.

• Relations between roots: If $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$, then

$$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}.$$

• Factor theorem: If $f(a)=0$, then $(x-a)$ is a factor of $f(x)$.
• Proportion rule: If $a:b=c:d$, the product of the extremes equals the product of the means:

$$ad=bc.$$ When the denominators are non-zero, $$\frac{a}{c}=\frac{b}{d}.$$

• Matrix addition and subtraction: These are possible only for matrices of the same order; corresponding elements are added or subtracted.
• Matrix multiplication: For matrices $A$ and $B$, $AB$ is possible only when the number of columns of $A$ equals the number of rows of $B$.
• Determinant of a $2\times 2$ matrix: For

$$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],\det =ad-bc.$$

• Inverse of a $2\times 2$ matrix:

$${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right],$$ provided $ad-bc\ne 0$.

• Arithmetic progression: If the first term is $a$, the common difference is $d$, and the $n$th term is $T_n$,

$$T_n=a+(n-1)d.$$ The sum of the first $n$ terms is $$S_n=\frac{n}{2}[2a+(n-1)d].$$

• Geometric progression: If the first term is $a$, the common ratio is $r$, and the $n$th term is $T_n$,

$$T_n=a{r}^{n-1}.$$ The sum of the first $n$ terms is $$S_n=\frac{a(r^n-1)}{r-1}\text{when }r\ne 1,$$ or equivalently $$S_n=\frac{a(1-r^n)}{1-r}.$$

• Distance between two points: For $(x_1,y_1)$ and $(x_2,y_2)$,

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

• Midpoint: The midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ is

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

• Internal section formula: If a point divides the segment joining $(x_1,y_1)$ and $(x_2,y_2)$ internally in the ratio $m:n$, its coordinates are

$$\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right).$$

• Slope: The slope of the line through two points is

$$\frac{y_2-y_1}{x_2-x_1},$$ provided $x_2\ne x_1$.

• Area of a triangle: For vertices $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$,

$$\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|.$$

Worked Methods

Solving a linear inequation

• Apply the same valid operation to both sides.
• Add or subtract terms as required.
• If multiplying or dividing by a negative number, reverse the inequality sign.
• State the resulting solution set.

The central method is consistent algebraic manipulation, while respecting the reversal condition for negative multiplication or division.

Solving a quadratic equation

A quadratic equation can be solved by factorisation, completing the square, or the quadratic formula.

Using the quadratic formula for $$a{x}^2+bx+c=0:$$

• Identify $a$, $b$, and $c$.
• Calculate the discriminant $D=b^2-4ac$.
• Substitute into

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

• Interpret the roots using the value of $D$.

Factorisation may be preferable when the quadratic has simple factors. Completing the square provides an alternative algebraic form. The appropriate method depends on the form of the equation.

Using the factor theorem

• Let the polynomial be $f(x)$.
• Evaluate $f(a)$.
• If $f(a)=0$, conclude that $(x-a)$ is a factor of $f(x)$.
• Use the factor to simplify or solve the polynomial expression.

Solving a proportion

For $$a:b=c:d:$$

• Rewrite the ratios as fractions if necessary.
• Cross-multiply:

$$ad=bc.$$

• Solve the resulting equation for the unknown quantity.
• Alternatively, use

$$\frac{a}{c}=\frac{b}{d}$$ when the denominators are non-zero.

Ratios and proportions are used in scale drawings, mixtures, rates, sharing, and similar figures.

Adding or subtracting matrices

• Check that the matrices have the same order.
• Add or subtract corresponding elements.
• Write the resulting matrix in the same order.

Addition or subtraction is not defined for matrices of different orders.

Multiplying matrices

• Check that the number of columns of the first matrix equals the number of rows of the second.
• Multiply each row of the first matrix by the corresponding column of the second.
• Add the products to obtain each entry of the product matrix.
• Arrange the results in a matrix whose order is determined by the remaining row and column counts.

Finding a determinant and inverse

For $$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]:$$

• Calculate the determinant:

$$ad-bc.$$

• If $ad-bc=0$, the inverse does not exist.
• If $ad-bc\ne 0$, interchange $a$ and $d$.
• Change the signs of $b$ and $c$.
• Multiply by $1/(ad-bc)$:

$$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right].$$

Finding terms and sums of an arithmetic progression

For an arithmetic progression with first term $a$, common difference $d$, and term number $n$:

• Use

$$T_n=a+(n-1)d$$ to find the $n$th term.

• Use

$$S_n=\frac{n}{2}[2a+(n-1)d]$$ to find the sum of the first $n$ terms.

Finding terms and sums of a geometric progression

For a geometric progression with first term $a$ and common ratio $r$:

• Use

$$T_n=a{r}^{n-1}$$ to find the $n$th term.

• If $r\ne 1$, use

$$S_n=\frac{a(r^n-1)}{r-1}.$$

• Equivalently, use

$$S_n=\frac{a(1-r^n)}{1-r}.$$

Solving coordinate-geometry problems

• Identify the coordinates of the relevant points.
• Select the required formula: distance, midpoint, section, slope, or triangle area.
• Substitute the coordinates carefully.
• Simplify the result and interpret it geometrically.

The principal formulas are: $$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2},$$ $$\text{midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right),$$ $$\text{section point}=\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right),$$ $$\text{slope}=\frac{y_2-y_1}{x_2-x_1},$$ and $$\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|.$$

Where It Goes Wrong

• Applying an operation to only one side of an equation or inequation invalidates the algebraic manipulation.
• Multiplying or dividing an inequation by a negative number without reversing the inequality sign gives an incorrect solution set.
• Using the quadratic formula without identifying the correct $a$, $b$, and $c$, or miscalculating $b^2-4ac$, produces incorrect roots.
• Matrix addition and subtraction are attempted for matrices of different orders, or matrix multiplication is attempted without checking that the columns of the first matrix match the rows of the second.
• The inverse of a $2\times 2$ matrix is calculated when $ad-bc=0$, although the inverse requires $ad-bc\ne 0$.
• The slope formula is used when $x_2=x_1$, even though the stated formula requires $x_2\ne x_1$; similarly, the alternative geometric-progression sum condition $r\ne 1$ must be observed.

What Gets Asked

This material supports questions requiring students to:

• define a linear inequation, solution set, quadratic equation, discriminant, ratio, proportion, factorisation, matrix, progression, coordinate geometry, distance formula, section formula, and slope;
• solve linear inequations and state their solution sets;
• solve quadratic equations by factorisation, completing the square, or the quadratic formula;
• determine the nature of quadratic roots using $D=b^2-4ac$;
• use the relationships $\alpha +\beta =-b/a$ and $\alpha \beta =c/a$;
• apply the factor theorem;
• solve ratio and proportion problems using $ad=bc$;
• add, subtract, multiply, find determinants of, and find inverses of matrices;
• calculate terms and sums in arithmetic and geometric progressions;
• find distances, midpoints, section points, slopes, and triangle areas from coordinates;
• apply algebra to scale drawings, mixtures, rates, sharing, similar figures, transformations, simultaneous linear equations, and regular numerical patterns.