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Linear Equations

2.0 Algebra · 2.2 Linear Equations

Syllabus tag: Kenya CBC | Grade 8 Mathematics | Strand 2.0 Algebra | Sub-Strand 2.2 Linear Equations (7 lessons)

Lesson objectives

By the end of this sub-strand, you should be able to:

  • Form linear equations in two unknowns in real-life situations
  • Solve linear equations in two unknowns by the substitution method
  • Solve linear equations in two unknowns by the elimination method
  • Apply linear equations in two unknowns in real-life situations

Linear Equations

One equation with two unknowns has many answers. Two equations with the same two unknowns usually have exactly one, and this topic finds it.

a) Forming the equations

Most questions arrive as words. Turning them into equations is the first and most important step.

Forming the equations Forming the equations 2 pens, 3 books = cost 260 the first purchase as an equation → 2x + 3y = 260 x a pen, y a book 1 pen, 2 books = cost 160 the second purchase as an equation → x + 2y = 160 two equations, two unknowns

Choose a letter for each unknown and say clearly what it stands for. Writing "x is the cost of a pen in shillings" prevents confusion later.

Then write one equation for each piece of information given.

You need as many equations as you have unknowns. Two unknowns need two equations.

b) The substitution method

Substitution makes one letter the subject in one equation, then replaces it in the other.

Substitution Substitution rearrange → x = 160 − 2y from the second equation substitute → 2(160 − 2y) + 3y = 260 put it in the first expand → 320 − 4y + 3y = 260 open the bracket simplify → 320 − y = 260 collect the y terms solve → y = 60 a book costs 60

Choose the easiest rearrangement. Here the second equation has a plain x, so making x the subject costs no fractions.

Once you have one value, put it back into either original equation to find the other. Here y = 60 gives x + 120 = 160, so x = 40.

c) The elimination method

Elimination adds or subtracts the equations so that one letter disappears.

Elimination Elimination first → 2x + 3y = 260 as it stands second × 2 → 2x + 4y = 320 match the x terms subtract → y = 60 the x terms cancel put y back → x + 120 = 160 into the second equation solve → x = 40 a pen costs 40

Multiply one or both equations so that one letter has the same coefficient in both.

Then subtract if the signs match, or add if they are opposite.

Here doubling the second equation gives 2x in both, so subtracting removes x entirely.

d) Which method to use

Substitution suits equations where a letter already stands alone, or has a coefficient of 1.

Elimination suits equations where the coefficients match already, or match after one small multiplication.

Both give the same answer. Neither is more correct.

e) Checking

Substitute both values into both original equations.

Take x = 40 and y = 60. The first gives 2(40) + 3(60) = 80 + 180 = 260. The second gives 40 + 2(60) = 160. Both are satisfied, so the answer is right.

Checking in only one equation proves nothing, because a wrong pair can still satisfy one of them.

f) Common errors

Forgetting to multiply every term when scaling an equation. If you double one side you must double all of it.

Subtracting wrongly when a negative is involved. Take the subtraction term by term.

Answering with one value only. A pair of equations in two unknowns needs both.

g) Where this is used

A trader works out two unit prices from two bulk purchases. A farmer splits land between two crops given a total area and a total cost. Any problem with two unknown quantities and two facts about them is this.

Words to know

  • Simultaneous equations -- a pair of equations involving the same two unknowns, solved together to find one solution that satisfies both.
  • Substitution method -- solving one equation for one unknown, then substituting that expression into the other equation.
  • Elimination method -- adding or subtracting two equations to cancel out one unknown.

:::checkpoint Check yourself

  1. Why do you need two equations to find two unknowns?
  2. Solve by substitution: x + y = 10 and x − y = 2.
  3. Solve by elimination: 3x + y = 14 and x + y = 6.
  4. Why should you check your answer in both original equations? :::

Bridge to practice

Try the exercises below -- on forming and solving linear equations in two unknowns using substitution and elimination.

Check yourselfPractise Linear Equations — 10 questions →Next in MathematicsArea →