6. Solve the following pairs of equations algebraically.
(i) 2a = 4 + 3b 3b − a = 4
(ii) p = 4 − 3q 3p = 2 + q
7. Solve the following pairs of equations simultaneously.
(i) x = 2y + 4 x = 3y – 1
(ii) a + 2β = 7 β + a = 4
(iii) 3a – 2b – 7=0 2a + 5b – 11 = 0
(iv) b = 5a + 5 b = 6a + 5
21.4 Creating and solving simultaneous equations
(iii) 5a = 3 + b 2a + b − 4 = 0
8. The path of a growing branch is described by the linear equation 2x + 3y = 15. A second branch growing beside the fi rst has a growth path described using the equation 4x − 3y = 3. Find the point where the branches cross.
9. The direction of one sword pointed in the air is described as 6x − y = 59 and the path of a second sword is described using the equation x + 15y = 25 . Find the value for x and the value for y where the two swords cross.
By the end of this section you should be able to: ● form simultaneous linear equations ● use simultaneous equations to solve a problem
We have already learned the steps of forming linear equations from mathematical problems and diff erent methods of solving simultaneous equations.
When we have a problem where we have to fi nd the values of two unknown quantities, we assume the two unknown quantities as x, y or any two of other algebraic symbols.
Then we form equations according to the given condition or conditions and solve the two equations simultaneously to fi nd the values of the two unknown quantities. Thus, we can work out the problem.
Solving word problems using simultaneous equations Before starting, read all the information in the question carefully and organise it into two sets of information.
1
Let x equal one unknown and let y equal the other unknown. Split the information into two separate sentences, each of which include both include x and y . Form an equation for each sentence.
2 3
Solve the two equations simultaneously.
Answer the question that is asked in the original question.
Worked example 1 The sum of two numbers is 14 and their diff erence is 2. Find the numbers.