Solving Equations and Straight-Line Graphs

Welcome. This is the first Mathematics block in your three-week prelim preparation plan. Today is a focused 45-minute revision session on two linked Year 11 Mathematics Standard 2 areas: Formulae and Equations and Linear Relationships.

By the end, you should be able to substitute safely, solve a linear equation while showing valid working, and read or plot the essential features of a straight-line graph. Most importantly, you will mark your work in a way that identifies why an answer went wrong, rather than just recording that it was wrong.

Have lined paper, graph paper or a ruled grid, a pencil, ruler, and your current Maths worksheet or textbook questions ready.

Time	Focus
3:45-3:49	Set up and recall the three methods
3:49-3:56	Read key ideas: substitution, equations, and lines
3:56-4:06	Work through substitution and equation methods
4:06-4:14	Read and plot straight-line graphs
4:14-4:25	Complete three selected questions from your school work
4:25-4:30	Mark, classify errors, and record the next step

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One lesson, three different jobs

These topics can look similar because they use letters, numbers, and equals signs. But they ask you to do different things.

If the question says...	Your job is...	Main idea
“Given $a=-3$, find...”	Substitute	Replace letters with the given values.
“Solve for $x$”	Solve an equation	Find the value that makes both sides equal.
“Graph” or “identify gradient/intercept”	Interpret a line	Connect the equation, slope, and coordinates.

Before starting any question, circle its instruction word: evaluate, solve, graph, find gradient, or find intercept. This takes only a few seconds, but it prevents using the wrong method.

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Substitution: replace first, calculate second

Substitution means replacing each pronumeral with its stated value. The safest habit is to write the expression again before calculating.

Suppose $p=-3$ and $q=4$, and you need to evaluate:

$$2p^2-3q$$

Write the substitution with brackets around a negative value:

$$2(-3{)}^2-3(4)$$

Now calculate in order:

$$2(9)-12=18-12=6$$

The answer is:

$$6$$

The bracket rule

If a negative number is substituted into a term with a power, brackets are essential:

$$(-3{)}^2=9$$

but

$$-3^2=-9$$

The first means “square negative three”; the second means “square three, then apply the negative sign.” In a prelim question, missing these brackets can change a correct method into an incorrect answer.

Use this compact substitution routine:

1. Copy the original expression.
2. Replace every letter with a number.
3. Put brackets around negative replacements.
4. Apply powers first, then multiplication or division, then addition or subtraction.
5. Check whether the sign of the final answer is reasonable.

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Solving equations: preserve balance

An equation states that two expressions have the same value. Think of the equals sign as a balanced scale: any operation performed on one side must also be performed on the other.

For example, solve:

$$5(2x-1)=35$$

First remove the factor of $5$ by dividing both sides by $5$:

$$\frac{5(2x-1)}{5}=\frac{35}{5}$$ $$2x-1=7$$

Then add $1$ to both sides:

$$2x=8$$

Finally, divide both sides by $2$:

$$x=4$$

A solution should be checked in the original equation:

$$5(2(4)-1)=5(7)=35$$

So $x=4$ is correct.

A reliable equation-solving structure

Avoid writing “move the number across” without showing what happened. Instead, write one clear algebraic step per line:

$$3x+8=26$$ $$3x=18$$ $$x=6$$

This layout makes your working easy to mark and makes errors easy to locate. In particular:

• undo addition or subtraction before undoing multiplication or division;
• when brackets are present, consider whether dividing first will simplify the equation;
• always check a final value by substituting it back into the original equation.

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Straight-line graphs: connect equation, slope, and intercept

A straight-line relationship is commonly written as:

$$y=mx+b$$

Here:

• $m$ is the gradient, or slope;
• $b$ is the $y$-intercept, where the line crosses the $y$-axis;
• the $y$-intercept has coordinates $(0,b)$.

The gradient tells you how much $y$ changes for each change in $x$:

$$m=\frac{\text{rise}}{\text{run}}$$

For a positive gradient, the line rises as you move left to right. For a negative gradient, it falls. A horizontal line has gradient $0$.

From the graph, use two neighbouring marked points, such as $(2,1)$ and $(3,4)$:

$$m=\frac{4-1}{3-2}=\frac{3}{1}=3$$

The line crosses the $y$-axis at $-5$, so $b=-5$. Its equation is therefore:

$$y=3x-5$$

For today, make sure you can read this information from a graph and use an equation to plot a graph. Next session, you will practise constructing straight-line equations more systematically.

Plotting from an equation

For a line such as:

$$y=-\frac{2}{3}x+4$$

1. Start with the $y$-intercept: $b=4$, so plot $(0,4)$.
2. Read the gradient:

$$-\frac{2}{3}$$

This means go down 2 and right 3.
3. From $(0,4)$, move down 2 and right 3 to plot $(3,2)$.
4. Use a ruler to draw a straight line through the points and extend it with arrows.

A quick check: because the gradient is negative, the line should slope downward from left to right. If yours slopes upward, check the sign.

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The 11-minute exam-style practice and marking routine

Use your current class worksheet, textbook revision questions, or teacher-provided questions. Choose only:

1. One substitution question, preferably involving a negative value, a fraction, or a power.
2. One linear equation to solve, preferably with brackets.
3. One straight-line graph question, where you either identify gradient and intercepts or plot from an equation.

Work independently for about 8 minutes. Show every line of working: prelim marks can be awarded for a sound method even if a later arithmetic slip occurs.

Then spend 3 minutes marking. Use worked answers, your notes, or teacher solutions only after you have finished all three questions.

Mark for method, not just final answer

For each question, use this checklist.

Substitution

• Did I replace every pronumeral?
• Did I use brackets around every negative value?
• Did I follow the order of operations?

Equation

• Did I apply the same operation to both sides?
• Is the variable isolated?
• Does my answer satisfy the original equation?

Straight-line graph

• Did I label axes and use a consistent scale?
• Did I plot the $y$-intercept as $(0,b)$?
• Did I interpret the gradient as rise over run, including its sign?
• Does the line pass through the plotted points?

If something is incorrect, record the error using one of these categories:

Error type	Example	Fix for next time
Sign error	Forgetting brackets around $-3$	Bracket every negative substitution.
Operation error	Dividing only one side of an equation	Write the operation on both sides.
Order error	Adding before evaluating a square	Complete powers before multiplication.
Graph-reading error	Treating $b$ as the $x$-intercept	Remember: $b$ is where $x=0$.
Presentation error	Points are unclear or line is not straight	Use a ruler and label coordinates.

Write one sentence in your error log, even if all three answers are correct. For example: “I solved equations correctly, but I need to show the division step on both sides more clearly.” This makes Friday's mixed revision much more targeted.

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Wrap-up

Today’s key distinction is:

• substitution replaces letters with known values;
• equation solving finds an unknown while keeping both sides balanced;
• linear graphs connect the equation $y=mx+b$ with gradient and intercept.

For future questions, begin by identifying which of those three jobs the instruction requires. Use brackets for negative substitutions, check every equation solution in the original equation, and treat a graph as evidence: the gradient and intercept must match the line you have drawn.

In the next Maths session, you will build from this foundation by rearranging formulae and constructing straight-line equations. Keep today’s error log; it will show whether you need to prioritise algebraic rearrangement, graph features, or sign accuracy.