L earn M aths MATHEMATICS For GCSE Solving Linear Equations Visit: www.maths4maths.co.uk For video lessons and other resources Denis Dalaba MSc, B.Ed.(Hons) www.maths4maths.co.uk by Denis Dalaba Page 1
Solving linear equations Some examples of linear equations are An equation is a mathematical statement stating that two expressions are equal. The unknown number is a particular number that can be worked out. To solve an equation means finding a. that makes the equation a true statement. This can be done by mental inspection or using algebraic manipulation. Solving equations by mental inspection or trial and improvement Very simple equations can often be solved mentally or by trial and improvement Example Solve the equations Solutions v i. x + 3 = 5 ii. 2n = 20 iii. 8 4 Stop the video and Try Exercise 1 www.maths4maths.co.uk by Denis Dalaba Page 2
Algebraic method We are now going to learn a systematic procedure (algebraic way) of solving equations. The algebraic method can be used to solve both simple and complex equations. Introduction Solving equations depend on two maths properties: a. the.. b. the.. a. = In the figure above, the left hand side (LHS) and right hand side (RHS) both have 10 rectangles each. www.maths4maths.co.uk by Denis Dalaba Page 3
i. Subtracting from both sides will result in both being left with... ii. Adding.. to both sides will result in both having... iii. Multiplying both sides by will result in both having iv. Dividing both sides by will result in both being left with. In algebra whenever we perform an operation on the left hand side (LHS) of an equal sign we must do the same to the right hand side (RHS) to maintain the equivalence (that is to balance the equation). b. Inverse operations (Reverse operation) or undo property Additive Inverse 1. The additive inverse of + 5 is.. 2. The additive inverse of - 9 is. Or 1. To undo add 5 we... 2. To undo subtract 9 we Multiplicative Inverse (reciprocal) 3. The multiplicative inverse of 7 is.. 4. The multiplicative inverse of 10 is... 3. To undo multiply by 7 we 4. 4. To undo divide by 10 we Note In algebra, x 5 is written as 5 x www.maths4maths.co.uk by Denis Dalaba Page 4
Fact 1 i. 3 + ( ) = ii. - 2 + = iii. a + ( ) = Fact 2 By the property of inverse (or reciprocals) we know that a. 5 = b. 7 1 = c. 2 9 = Solving equations We solve equations by reducing (restating) the equation to an equivalent equation of the form x = a where a is an integer. That is x = some number. Example 2x + 5 = 17 We can work out mentally that x = 6 www.maths4maths.co.uk by Denis Dalaba Page 5
Solving equations involving additive inverse We are now going to apply additive inverse with the balancing property to solve equations Example Solve the equations i. x + 3 = 7 ii. y 9 = 21 Solution Stop the video and Try Exercise 2 www.maths4maths.co.uk by Denis Dalaba Page 6
Solving equations using multiplicative inverse We are now going to apply multiplicative inverse (also called the reciprocal) with the balancing property to solve equations. Type 1- Multiplication Solve the equations i. 15m = 45 ii. 12b = 6 iii. 10 = 5t Strategy: To solve an equation in which the variable is multiplied by a constant, we start by multiplying both the LHS and the RHS by the reciprocal of the constant. Stop the video and Try Exercise 3 www.maths4maths.co.uk by Denis Dalaba Page 7
Type 2 Division i. Examples Solve the equations x = 20 ii. 1 c = 2 iii. 1 4 9 0.5 y = 16 Stop the video and Try Exercise 4 www.maths4maths.co.uk by Denis Dalaba Page 8
Type 3- Multiplication & Division Example Solve i. 2 3 y = 12 Stop the video and Try Exercise 5 www.maths4maths.co.uk by Denis Dalaba Page 9
Solving equations using combination of strategies We are now going to use a combination of strategies (add, subtract, multiply and or divide) to solve equations. Examples Solve the following a. 5 x + 3 = 18 b. 4w 7 = 21 c. 2 7 y + 1 = 11 Solution Stop the video and Try Exercise 6 www.maths4maths.co.uk by Denis Dalaba Page 10
Solving equations involving brackets (parentheses) Example Solve the following i. 2(x + 5) = 21 ii. 4(y +2) = 20 iii. 7 (x - 1 ) = 12 iv. 5(2n + 7 ) + 3 = 78 Strategy: Expand to get rid of the parentheses(brackets) Group and simplify like terms Do the relevant inverse operations Note: In some cases expanding the bracket first in not necessary 500 (2x - 1) = 1000 Start by dividing by the HCF Stop the video and Try Exercise 7 www.maths4maths.co.uk by Denis Dalaba Page 11
Solving equations involving fractions Examples Solve the following 2 5 3 7 i. x 4 ii. x 5 ii. iii. 3t 9 11 4 iv. 7 6m 3 5 Stop the video and Try Exercise 8 www.maths4maths.co.uk by Denis Dalaba Page 12
Solving equations with the variable (unknown) on both the LHS and RHS Examples Solve the equations i. 7x + 1 = x + 13 ii. 5b + 1 = 16-2b ii. 5 + 3y = 2y -7 Stop the video and Try Exercise 9 www.maths4maths.co.uk by Denis Dalaba Page 13
Solving equations containing exponentials and radicals Examples i. x 2 9 ii. 2y 2 = 32 iii. 3x 5 Stop the video and Try Exercise 10 www.maths4maths.co.uk by Denis Dalaba Page 14
Mixed questions Find x? 2x 26 0 x www.maths4maths.co.uk by Denis Dalaba Page 15