Creating Linear Equations From Information

One area which commonly students find challenging in GCSE Maths, is how to create Linear equations from Exam Question Paper Information, so that they can solve them.

Here is an example of this- taken from the new 9-1 syllabus: AQA Higher Tier, Set 2 Paper 2, Question 13:

The 3 sides of a triangle are as follows:

AB= 3X -9, AC= 2X+4, and BC = 5X-11. All lengths are in centimetres.

Given that AB:BC = 1:2, show that:

AC:BC=3:4.

Solution: The key to solving this is firstly understanding the principle of ratios:

If AC: BC = 1:2, this tells us that BC is twice the length of AB.

So, 2 (3X-9) = 5X-11.

If we now expand the bracket on the left-hand side, we get:

6X-18= 5X-11.

Then if we subtract the smaller X term, which is 5X, on both sides of the equation, we are left with:

X-18= -11.

Then if we add 18 to both sides, to remove the -18 from the left-hand side, we are left with:

X=7.

Now using substitution of X=7, into AC, we get: 2X+4 = 2 x 7 +4= 18.

Also, substituting X=7 into BC, we get : 5X-11, 5 x 7 -11 = 24.

So, AC= 18, and BC= 24.

So, AC:BC = 18:24, or dividing both sides by a common factor of 6, we get:

AC: BC= 3:4.

Another common example is when you have Algebra in a Geometry question, such as:

The 3 angles of a triangle (in degrees) are:

X + 30, 2X +10, and X +40.

Find the largest angle in the triangle.

Solution:

In order to find the largest angle, we must find the value of X.

To achieve this, we must use the rule we know about triangles: Angles in a Triangle add up to 180 degrees. So we create an equation, by summing the angles in a triangle in terms of the algebra on the left-hand side, and by making the right-hand side equal to 180 degrees:

So, X+30 + 2X+10+X+40 = 180

Next, simplify the left-hand side of the equation by collecting the X terms separately and the number terms separately.

So, we get: 4X+80= 180.

Them subtract 80 from both sides to remove the number term from the left-hand side, we get:

4X= 100.

Dividing by 4, we then get:

X=25.

Then substitute back into all 3 angle expressions, we get:

X+30= 25+30= 55 degrees.

2X+10= 2 x 25 +10 = 60 degrees

X+40 = 25+40= 65 degrees.

So, the largest angle in the triangle = 65 degrees.