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Substitution
Substitution Revision
Substitution
Substitution means replacing letters or symbols in algebraic equations or expressions with numbers to find out their total value.
Using Substitution
Substitution can be helpful in utilising formulae and equations.
For example, we know
\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}
To utilise this helpful equation, we need to use substitution:
It takes Esme 45\text{ minutes} to travel to school, which is 3\text{ km} away. We can use substitution into the equation above the find the speed Esme travels in \text{km/h}:
Distance = 3\text{ km}
Time = 0.75\text{ hours}
\text{Speed} = \dfrac{3}{0.75}
\text{Speed} = 4\text{ km/h}
Example 1
Given
k = 5a + 19b - 12c - \dfrac{d}{4}
Work out the value of k when
a=8\\ b=2\\ c=4\\ d=48[2 marks]
Substituting the values of a, b, c, and d into the equation:
k = 5\times8 + 19\times2 - 12\times4 - \dfrac{48}{4}
You must remember BIDMAS when using substitution, so division/multiplication must come before addition/subtraction.
k = 40 + 38 - 48 - 12
k =18
Example 2
Using the quadratic formula is an example of substitution.
Given the quadratic formula
x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}
Solves equations in the form
ax^2+bx+c=0
Use substitution into the quadratic formula to solve
3x^2+5x+2=0
[3 marks]
We can see from the equation that:
a=3\\ b=5\\ c=2So, we can substitute these into the quadratic equation:
x=\dfrac{-5\pm\sqrt{5^2-4(3)(2)}}{2(3)}
Which gives us the solutions x=-\dfrac{2}{3} and x=-1
Substitution Example Questions
Question 1:
Z=2a+5b^3
Find the value of Z when a=36 and b=\dfrac{1}{2}a.
[2 marks]
If a=36, b=\dfrac{1}{2}(36) = 18
Substitution:
Z=2(36)+5(18)^3\\ Z=29232
Question 2:
P = 5m + 16
P = 46
Work out the value of m
[2 marks]
Substitution:
46 = 5m + 16Rearrange and solve:
46-16=5m\\ 30=5m\\ m=6Question 3: A teacher creates an equation to estimate the English exam scores of her students based on their Science scores.
y=2x-21
Where x is their Science score, and y is the estimate for the English score.
Ezra got 78 in his Science exam. Predict his English result.
[2 marks]
Substitution:
y = 2(78)-21\\ y = 135Ezra would be expected to score 135 in his English test.
