Year 12 Maths - Methods (Unit 3 & Unit 4) Algebra and equations

Solution of literal equations and systems of equations

Q: What is a literal equation?

A literal equation is an equation whose coefficients and constants are pronumerals (letters) rather than fixed numbers, such as a x plus b equals c. Solving it means rearranging to make one chosen variable the subject in terms of the other letters, for example x equals (c minus b) over a, valid provided a is not zero.

Q: How do you solve a literal equation for a variable?

Treat the target variable like the unknown and every other letter like a known constant. Collect all terms containing that variable on one side, factor it out, then divide by its coefficient. Note any letter you divide by must be non-zero, so state that condition.

Q: What are the methods for solving simultaneous linear equations?

The two standard methods are substitution and elimination. In substitution you make one variable the subject of one equation and substitute it into the other. In elimination you add or subtract multiples of the equations to cancel one variable. Both give the point where the two lines meet.

Q: When does a pair of simultaneous linear equations have no solution?

When the two lines are parallel but not the same line, they never meet, so there is no solution. This happens when the equations have equal gradients but different intercepts, for example 3x plus y equals 3 and 3x plus y equals 4.

Q: When does a system have infinitely many solutions?

When the two equations describe the same line, they are coincident, so every point on the line is a solution and there are infinitely many. This happens when one equation is a constant multiple of the other, so the coefficients and the constant are all in the same ratio.

Q: How do you find a parameter value that gives a unique solution?

A system a1 x plus b1 y equals c1 and a2 x plus b2 y equals c2 has a unique solution exactly when a1 b2 minus a2 b1 is not zero, which means the two lines have different gradients. Set that expression not equal to zero and solve for the parameter; the value that makes it zero is the one to exclude.

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Theory

A literal equation has pronumerals for its coefficients, and is solved by making one variable the subject in terms of the others. A pair of simultaneous linear equations is solved by substitution or elimination, and geometrically the two lines either meet once (a unique solution), are parallel (no solution), or coincide (infinitely many solutions).

A literal equation is one in which the coefficients and constants are letters (parameters) rather than numbers, such as \(ax+b=c\). To solve it for a chosen variable you treat that variable as the unknown and every other letter as a known constant: collect its terms on one side, factorise it out, then divide by its coefficient. For example \(ax+b=c\) gives \(x=\dfrac{c-b}{a}\), valid provided \(a\neq 0\).

A pair of simultaneous linear equations such as \(a_1x+b_1y=c_1\) and \(a_2x+b_2y=c_2\) asks for the \((x,y)\) that satisfies both at once. Two standard methods are used: substitution, where one variable is made the subject of one equation and put into the other, and elimination, where the equations are added or subtracted (after scaling) to cancel one variable.

Each linear equation is a straight line, so the solution is the point of intersection. This gives three geometric possibilities: the lines intersect once (a unique solution), are parallel and distinct (no solution), or are coincident — the same line (infinitely many solutions). Equal gradients with different intercepts give no solution; identical equations (one a multiple of the other) give infinitely many.

Key idea. A system \(a_1x+b_1y=c_1,\ a_2x+b_2y=c_2\) has a unique solution exactly when \(a_1b_2-a_2b_1\neq 0\) (different gradients). When \(a_1b_2-a_2b_1=0\) the lines are parallel: no solution if the constants break the pattern, infinitely many if the whole equation is a multiple of the other.

Unique solution — the lines meet once, here at \((1,1)\)

No solution — parallel lines (equal gradient, different intercept) never meet

Infinitely many solutions — coincident lines: one equation is a multiple of the other

A literal equation is rearranged by isolating the target variable; if it appears on both sides, gather and factorise:

\[ax+b=cx+d \quad\Rightarrow\quad x=\dfrac{d-b}{a-c}\quad(a\neq c)\]

x = \frac{d - b}{a - c}

The general pair of simultaneous linear equations:

\[a_1x+b_1y=c_1 \qquad a_2x+b_2y=c_2\]

a_{1} x + b_{1} y = c_{1}

The number of solutions is decided by the coefficients — a unique solution requires different gradients:

\[\text{unique} \iff a_1b_2-a_2b_1\neq 0\]

a_{1} b_{2} - a_{2} b_{1} \neq 0

When the lines are parallel, the ratios of the coefficients separate the two remaining cases:

\[\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}\ \text{(none)} \qquad \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\ \text{(infinitely many)}\]

\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}

Geometric reading. Different gradients \(\Rightarrow\) one crossing point (unique). Same gradient, different intercept \(\Rightarrow\) parallel, no solution. Same line \(\Rightarrow\) infinitely many solutions.

Solving literal equations and simultaneous systems

Literal equation. Decide which letter is the subject; move every term containing it to one side and everything else to the other.
Factorise and divide. Factor the target variable out of its terms, then divide by the resulting coefficient — state any letter you divide by is non-zero.
System by substitution. Make one variable the subject of one equation, substitute into the other, solve the single-variable equation, then back-substitute for the second variable.
System by elimination. Scale the equations so one variable has matching coefficients, add or subtract to eliminate it, solve, then back-substitute.
Classify the solutions. Compare gradients: different gradients give a unique point; equal gradients give parallel lines — no solution if the intercepts differ, infinitely many if the equations are multiples.

Choosing a method. Use substitution when a variable already has coefficient \(1\) (easy to isolate); use elimination when the coefficients line up neatly for adding or subtracting.

Example 1 — literal equation

Make \(x\) the subject of \(ax+b=c\).

Solution

Isolate the \(x\)-term — subtract \(b\) from both sides:

\(ax+b\)	\(=\)	\(c\)
\(ax\)	\(=\)	\(c-b\)

Divide by the coefficient \(a\) (with \(a\neq 0\)):

\(x\)

\(=\)

\(\dfrac{c-b}{a}\)

\(\therefore\ x=\dfrac{c-b}{a}\quad(a\neq 0)\)

x = \frac{c - b}{a}

Example 2 — substitution

Solve \(y=2x-1\) and \(3x+y=9\).

Solution

Substitute \(y=2x-1\) into the second equation:

\(3x+(2x-1)\)	\(=\)	\(9\)
\(5x-1\)	\(=\)	\(9\)
\(5x\)	\(=\)	\(10\)
\(x\)	\(=\)	\(2\)

Back-substitute \(x=2\) into \(y=2x-1\):

\(y\)

\(=\)

\(2(2)-1=3\)

\(\therefore\ (x,y)=(2,3)\)

(x, y) = (2, 3)

Example 3 — elimination

Solve \(2x+y=5\) and \(x-y=1\), and interpret the result.

Solution

Add the equations — the \(y\) terms cancel:

\((2x+y)+(x-y)\)	\(=\)	\(5+1\)
\(3x\)	\(=\)	\(6\)
\(x\)	\(=\)	\(2\)

Back-substitute \(x=2\) into \(x-y=1\):

\(2-y\)	\(=\)	\(1\)
\(y\)	\(=\)	\(1\)

\(\therefore\ (x,y)=(2,1)\) — the lines cross once (unique solution)

(x, y) = (2, 1)

Example 4 — parameter and number of solutions

For the system \(kx+2y=6\) and \(3x+y=4\), find the values of \(k\) for which there is a unique solution, and describe what happens otherwise.

Solution

Unique solution requires \(a_1b_2-a_2b_1\neq 0\) with \(a_1=k,\ b_1=2,\ a_2=3,\ b_2=1\):

\(a_1b_2-a_2b_1\)	\(=\)	\(k(1)-3(2)\)
	\(=\)	\(k-6\)

Set this non-zero for a unique solution:

\(k-6\)	\(\neq\)	\(0\)
\(k\)	\(\neq\)	\(6\)

Check the exceptional value \(k=6\) — the equations become \(6x+2y=6\), i.e. \(3x+y=3\), and \(3x+y=4\):

\(3x+y\)	\(=\)	\(3\)
\(3x+y\)	\(=\)	\(4\)

\(\therefore\) unique solution for \(k\neq 6\); at \(k=6\) the lines are parallel and distinct, so there is no solution

k \neq 6

Common pitfalls

Dividing by a pronumeral that might be zero. When you divide both sides of a literal equation by a letter, that letter must be non-zero. Solving \(ax=c-b\) gives \(x=\dfrac{c-b}{a}\) only if \(a\neq 0\) — always state the condition.

Sign slips in elimination. When you subtract one equation from another, subtract every term, including the constant on the right. Lining the equations up in columns before subtracting avoids losing a sign.

Equal gradients do not always mean "no solution". Parallel lines with different intercepts give no solution, but if the two equations are the same line (one a multiple of the other) there are infinitely many. Check the constants before deciding.

Frequently asked questions

What is a literal equation?

An equation whose coefficients and constants are pronumerals rather than numbers, such as \(ax+b=c\). Solving it means making one variable the subject in terms of the others, e.g. \(x=\dfrac{c-b}{a}\) provided \(a\neq 0\).

How do you solve a literal equation for a variable?

Treat the target variable as the unknown and every other letter as a constant. Gather its terms on one side, factorise it out, then divide by its coefficient — noting any letter you divide by must be non-zero.

What are the methods for solving simultaneous linear equations?

Substitution — make one variable the subject and substitute into the other equation; and elimination — add or subtract (scaled) equations to cancel a variable. Both find the point where the lines meet.

When does a pair of simultaneous linear equations have no solution?

When the lines are parallel but distinct (equal gradient, different intercept) they never meet, so there is no solution — for example \(3x+y=3\) and \(3x+y=4\).

When does a system have infinitely many solutions?

When the two equations are the same line (one is a multiple of the other), so all coefficients and the constant share a ratio. Every point on the line satisfies both equations.

How do you find a parameter value that gives a unique solution?

A unique solution occurs when \(a_1b_2-a_2b_1\neq 0\). Set that expression not equal to zero and solve for the parameter; the value making it zero is the one to exclude, then test it to see whether it gives no solution or infinitely many.

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Solution of literal equations and systems of equations

📖 Theory

Solving literal equations and simultaneous systems

Common pitfalls

Frequently asked questions

Theory