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Maths 9 โ€“ Equations and systems of equations

๐Ÿ“˜ Notes

Equations and Systems of Equations

In these questions, you need to solve linear equations, quadratic equations, equations written as a product of factors, and systems of two linear equations.

Sometimes the question does not ask for all roots directly. It may ask for the larger root, the smaller root, the sum of the roots, or the product \(x \cdot y\).

1. Linear equations

To solve a linear equation, collect all terms with \(x\) on one side and numbers on the other side.

\[ ax+b=cx+d \]

Useful rules:

  • If there is a minus before brackets, change all signs inside the brackets.
  • If there is a number outside brackets, multiply everything inside by that number.

2. Quadratic equations

First write the equation in the form

\[ ax^2+bx+c=0. \]

Then factorise if possible.

\[ x^2+x=12 \quad \Longrightarrow \quad x^2+x-12=0 \]
\[ (x+4)(x-3)=0 \]

Use the zero product rule: if a product is zero, then at least one factor must be zero.

3. Equations written as a product

If

\[ (x-3)(3x+15)(2x-4)=0, \]

then solve each factor separately:

\[ x-3=0 \] \[ 3x+15=0 \] \[ 2x-4=0 \]

Then collect all roots and answer exactly what the question asks for.

4. Systems of two equations

A system such as

\[ \begin{cases} x+y=2 \\ 2x-y=-4 \end{cases} \]

can be solved by elimination or substitution.

Elimination: add or subtract the equations to remove one variable.

Substitution: make one variable the subject in one equation and substitute into the other.

5. What the question may ask for

If the question asks for What to do
the larger root Find both roots and choose the greater one
the smaller root Find both roots and choose the smaller one
the sum of the roots Add the roots
the product \(x \cdot y\) First solve the system, then multiply \(x\) and \(y\)
the sum \(x+y\) First solve the system, then add \(x\) and \(y\)

6. Worked examples

Example 1: Quadratic equation

Solve \(x^2+x=12\). Give the larger root.

Solution

\[ x^2+x-12=0 \] \[ (x+4)(x-3)=0 \]

So

\[ x=-4 \quad \text{or} \quad x=3 \]
The larger root is \(3\).

Example 2: Linear equation with brackets

Solve \(-(x+6)=2x+18\).

Solution

\[ -x-6=2x+18 \] \[ -3x=24 \] \[ x=-8 \]
Answer: \(x=-8\)

Example 3: System of equations

Solve the system. Give the product \(x \cdot y\):

\[ \begin{cases} x+y=2 \\ 2x-y=-4 \end{cases} \]

Solution

Add the equations:

\[ (x+y)+(2x-y)=2+(-4) \] \[ 3x=-2 \] \[ x=-\frac{2}{3} \]

Now find \(y\):

\[ x+y=2 \] \[ -\frac{2}{3}+y=2 \] \[ y=\frac{8}{3} \]

Now multiply:

\[ x \cdot y= -\frac{2}{3}\cdot \frac{8}{3} =-\frac{16}{9} \]
Answer: \(-\dfrac{16}{9}\)

Example 4: Product of factors

Solve \((x-3)(3x+15)(2x-4)=0\). Give the sum of the roots.

Solution

\[ x-3=0 \Rightarrow x=3 \] \[ 3x+15=0 \Rightarrow x=-5 \] \[ 2x-4=0 \Rightarrow x=2 \]

Sum of roots:

\[ 3+(-5)+2=0 \]
Answer: \(0\)

Example 5: Another quadratic equation

Solve \(x^2=2x+24\). Give the smaller root.

Solution

\[ x^2-2x-24=0 \] \[ (x-6)(x+4)=0 \]

So

\[ x=6 \quad \text{or} \quad x=-4 \]
The smaller root is \(-4\).

Example 6: System and sum \(x+y\)

Solve the system. Give \(x+y\):

\[ \begin{cases} x-y=3 \\ 2x+y=12 \end{cases} \]

Solution

Add the equations:

\[ (x-y)+(2x+y)=3+12 \] \[ 3x=15 \] \[ x=5 \]

Now find \(y\):

\[ x-y=3 \] \[ 5-y=3 \] \[ y=2 \]

Then

\[ x+y=5+2=7 \]
Answer: \(7\)

7. Common mistakes

  • Not moving all terms to one side in quadratic equations.
  • Changing signs incorrectly when opening brackets with a minus.
  • Stopping after finding the roots, when the question asks for the larger root, the smaller root, or the sum.
  • In a system, finding \(x\) but forgetting to find \(y\).
  • Making arithmetic mistakes when adding or multiplying the final answers.

8. Summary

\[ ax+b=cx+d \] \[ ax^2+bx+c=0 \] \[ \text{If } abc=0, \text{ then at least one factor is } 0 \]

In these questions, the main idea is to solve carefully and then give exactly the value the question asks for.