Substitution Revision Notes for Grade 11 NSC Matric Mathematics

Substitution

What is substitution?

Substitution is a powerful algebraic technique used to simplify complex equations by replacing a repeated expression with a single variable. This method is particularly useful when working with equations that contain the same expression multiple times, as it transforms a complicated equation into a simpler form that is much easier to solve.

When you encounter an equation where the same algebraic expression appears in different places, substitution allows you to temporarily replace that expression with a new variable, solve the resulting simpler equation, and then work backwards to find the original variable's value.

When to use substitution

Look for repeated expressions in your equation. These are identical algebraic terms that appear more than once. Substitution is most effective when:

The same expression appears at least twice in the equation
The repeated expression is reasonably complex (like $x^2 - 2x$ )
Direct solving would be difficult or time-consuming
The equation involves fractions where the repeated expression appears in denominators

infoNote

Substitution is particularly valuable when dealing with complex rational equations or equations where the same expression appears both as a base term and within fractions.

Step-by-step method for substitution

Follow these essential steps to solve equations using substitution:

Step 1: Identify any restrictions on the original variable

Find values that would make denominators equal to zero
These values cannot be part of your final answer

Step 2: Choose your substitution variable

Let a new variable (like $k$ ) equal the repeated expression
Rewrite the equation using this new variable

Step 3: Determine restrictions for your new variable

Check what values would make denominators zero in the new equation

Step 4: Solve the simplified equation

This often becomes a quadratic equation that you can factor or use the quadratic formula to solve

Step 5: Substitute back to find the original variable

Replace your substitution variable with the repeated expression
Solve each resulting equation

Step 6: Check your answers against all restrictions

Ensure none of your solutions violate the original restrictions

chatImportant

Critical: Always identify restrictions at the beginning of the problem. Forgetting this step is one of the most common mistakes students make, leading to invalid solutions that seem mathematically correct but violate the original equation's domain.

Worked example

lightbulbExample

Worked Example: Solving a Rational Equation with Substitution

Question: Solve for $x$ : $x^2 - 2x - \frac{3}{x^2 - 2x} = 2$

Solution:

Step 1: Determine restrictions for $x$

The denominator $x^2 - 2x$ cannot equal zero, so:

$x^2 - 2x = 0$
$x(x - 2) = 0$

Therefore $x \neq 0$ and $x \neq 2$

Step 2: Make the substitution

Notice that $x^2 - 2x$ is repeated. Let $k = x^2 - 2x$

The equation becomes: $k - \frac{3}{k} = 2$

Step 3: Find restrictions for $k$

Since we have $\frac{3}{k}$ in our equation, we need $k \neq 0$

Step 4: Solve for $k$

$k - \frac{3}{k} = 2$

Multiply through by $k$ :

$k^2 - 3 = 2k$
$k^2 - 2k - 3 = 0$
$(k + 1)(k - 3) = 0$

Therefore $k = -1$ or $k = 3$

Both values satisfy $k \neq 0$ , so both are valid.

Step 5: Substitute back to find $x$

For $k = -1$ :

$x^2 - 2x = -1$
$x^2 - 2x + 1 = 0$
$(x - 1)(x - 1) = 0$

Therefore $x = 1$

For $k = 3$ :

$x^2 - 2x = 3$
$x^2 - 2x - 3 = 0$
$(x + 1)(x - 3) = 0$

Therefore $x = -1$ or $x = 3$

Step 6: Check final answers

All three values ( $x = -1$ , $x = 1$ , and $x = 3$ ) satisfy our original restrictions ( $x \neq 0$ and $x \neq 2$ ).

Final answer: $x = -1$ , $x = 1$ , or $x = 3$

Key exam tips

infoNote

Essential Strategies for Success

Always check restrictions twice - once for the original variable and once for your substitution variable
Work systematically through each step rather than trying to take shortcuts
Verify your final answers by substituting back into the original equation if time permits
Look for common repeated expressions like $(x + a)$ , $(x^2 + bx + c)$ , or expressions in denominators
Choose simple letters for substitution variables (like $k$ , $u$ , or $t$ ) to avoid confusion

bookmarkSummary

Key Points to Remember:

Substitution simplifies equations by replacing repeated expressions with single variables
Always identify restrictions first to avoid invalid solutions
Check restrictions at every stage - for both the original variable and substitution variable
Factor or use the quadratic formula to solve the simplified equation
Substitute back carefully and verify all solutions against the original restrictions

Substitution (Grade 11 NSC Matric Mathematics): Revision Notes

Substitution

What is substitution?

When to use substitution

Step-by-step method for substitution

Worked example

Key exam tips

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