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S1 · 5 Linear Equations

Secondary 1 · Chapter 5

Linear Equations

Sort expressions from equations, see why a balance tilts, then solve linear equations on a balance — each step you take is written beside it in algebra, like the textbook’s tables — and link a bar model to the algebra of a word problem.

Read the notesSyllabus N7 7.1–7.4

Syllabus outcomes N7 7.1–7.4 (Equations and inequalities)
  1. 7.1 concept of equation
  2. 7.2 solving linear equations in one variable
  3. 7.3 solving simple fractional equations that can be reduced to linear equations such as x3 + x − 24 = 3; 3x − 2 = 6 (in the notes — not on this page)
  4. 7.4 formulating a linear equation in one variable to solve problems

Class Discussion Expressions and equations think! Mathematics 1A, p124

  1. Some expressions and equations are shown below. Identify which are expressions and which are equations. Then explain the difference between an expression and an equation.

Drag each card into Expressions or Equations — or tap a card, then tap Move … here under the box you want. Then press Check my sort.

  1. Which of the above equations are linear equations? Explain.

Tap each linear equation to mark it, then press Check.

Explore The balance think! Mathematics 1A, p125, Fig. 5.1, 5.2

The ‘=’ sign in an equation means that the expression on the LHS of the equation must have the same value as the expression on the RHS of the equation. We will explore this concept further using the idea of a balancing scale (or “balance” for short).

In Fig. 5.1, the left pan of the balance contains the expression 2 + 3, while the right pan contains one number 5. Since 2 + 3 = 5, the balancing scale is balanced. We say that the two expressions 2 + 3 and 5 are equivalent.

Try it — these steps are Math Lab’s, not the textbook’s.

  1. Press Add 1 to the left pan. What happens to the balance? What is written above the pans now?
  2. What must you do to the right pan to make the balance level again? Do it, then press Check.

The balance starts as in the textbook’s Fig. 5.1 (p125): 2 + 3 on the left pan and 5 on the right pan.

Investigation Solving linear equations — Part 1 think! Mathematics 1A, p125–126, Table 5.1

We will now explore how to solve equations of the form x + b = d, where b and d are constants. To find the value of x that satisfies the equation, we need to manipulate the equation in order to isolate x on one side of the equation.

Part 1: x + b = d, where b and d are constants

Example: x + 2 = 5

How to use the balance (Math Lab’s instructions)

  • Take away: tap the discs you want on ONE pan (a ring shows they are chosen), then press Take away … from both pans. The same discs are crossed out on the other pan too.
  • Add: tap a disc above the balance, choose how many (×), then press Add … to both pans.
  • Simplify both sides clears the crossed-out discs and the zero pairs. Divide both sides by splits each pan into equal groups.
  • Dragging a disc onto one pan, or off one pan, changes one side only — watch the balance tilt. Undo takes back the last step.
  • Check tells you whether the equation is solved. Stuck? Hint suggests the next step.
  • With a keyboard: Tab to a disc above the balance and press Enter to choose it, type how many in the box, then press Add … to both pans. To take discs away, add their opposite instead — subtracting 2 is the same as adding −2, as Table 5.3 does.

Take away 2 of the 1 discs from each pan, then simplify. Every step you take is written in the table under the balance — compare it with the textbook’s Table 5.1.

  1. Solve each of the following equations using the idea of the balancing scale.

Hint: When no discs can be taken away, what can we add to both sides of the equation to obtain zero pairs?

Investigation Solving linear equations — Part 2 think! Mathematics 1A, p126–127, Table 5.2

Part 2: ax + b = d, where a, b and d are constants

Example: 2x − 3 = 5

Make a zero pair with the −3 disc, simplify, then share each pan into 2 equal groups with Divide both sides by 2 and simplify again. Compare your table with Table 5.2.

  1. Solve each of the following equations using the idea of the balancing scale.

Hint: For part (c), you can divide by a negative number at an appropriate step.

  1. For the equation 2x − 5 = 7 in Question 2(a), can you divide both sides of the equation by 2 straightaway? Explain.

Try it on the balance for 2(a): before any other step, press Divide both sides by with 2 in the box. What happens on the balance, and why? Then try it in algebra: what do you get when you divide both sides of 2x − 5 = 7 by 2?

Investigation Solving linear equations — Part 3 think! Mathematics 1A, p127, Table 5.3

Part 3: ax + b = cx + d, where a, b, c and d are constants

Example: 5x + 4 = 2x − 8

Start by taking away the same number of x discs from each pan. When you cannot take a disc away from one side, add its opposite to both pans instead. Compare your table with Table 5.3.

  1. Solve each of the following equations using the idea of the balancing scale.

Hint: For part (d), you cannot take away 5 discs from the RHS. What can you add to both sides of the equation instead?

Practice Worked Example 1: Solving linear equations think! Mathematics 1A, p128

You have solved equations of all three forms on the balance. Open the box below to read what the Investigation showed, then solve the Worked Example on a balance.

Solve each of the following equations.

  1. 2x + 3 = 9

Part (b) of this Worked Example, 2(y − 4) = 5y − 3, starts by expanding the brackets with the Distributive Law — a step the balance cannot show — so it is not solved on a balance here. The notes work an equation with brackets like it, step by step.

Explore Linking models to algebra think! Mathematics 1A, p133, Table 5.4

Consider the following problem:

A man is 3 times as old as his son. In 10 years’ time, the sum of their ages will be 76. How old was the man when his son was born?

Build the model in the Model column, as in the textbook’s Table 5.4. Tap a box to change what it says (type 10 for a box of 10), use + and − for more or fewer boxes, and for the total tap the names of the rows, type the total and press Brace the selected rows. The Algebra column shows the textbook’s algebra one step at a time — compare each step with your model.

Model

Their ages now

In 10 years’ time,

Algebra

    Now answer the problem, using your model or the algebra.