Summary Revision Notes for Grade 10 NSC Matric Mathematics

Summary

This comprehensive summary covers the key concepts you need to master in equations and inequalities for your NSC Mathematics exam.

Types of equations

Linear equations

A linear equation is an equation where the variable has an exponent of 1. These equations form straight lines when graphed and have at most one solution.

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Key characteristics of linear equations:

The highest power of the variable is 1
They can be written in the form $ax + b = c$
They have exactly one solution (unless they're inconsistent)

Quadratic equations

A quadratic equation is an equation where the highest exponent of the variable is 2. These equations have at most two solutions.

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Key characteristics of quadratic equations:

The highest power of the variable is 2
They can be written in the form $ax^2 + bx + c = 0$
They can have 0, 1, or 2 real solutions

Systems of simultaneous equations

When you need to find the values of two unknown variables, you require two equations. These are called systems of simultaneous equations.

There are two main methods to solve them:

Algebraic solutions

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Two algebraic methods:

Substitution method: Solve one equation for one variable, then substitute into the other equation
Elimination method: Add or subtract equations to eliminate one variable

Graphical solutions

Draw the graph of each equation
The solution is the coordinates where the lines intersect
This gives you the $(x, y)$ values that satisfy both equations

Special types of equations

Literal equations

Literal equations contain several letters and variables. You solve for one variable in terms of the others.

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Worked Example: Literal Equation

If $A = \pi r^2$ , solve for $r$ in terms of $A$ and $\pi$ .

Solution: $A = \pi r^2$ $\frac{A}{\pi} = r^2$ $r = \sqrt{\frac{A}{\pi}}$

Word problems

Word problems require you to set up equations that represent the problem mathematically. The key is identifying what the variables represent and translating the words into mathematical expressions.

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Strategy for word problems:

Define your variables clearly
Identify what relationships exist between quantities
Translate word descriptions into mathematical expressions
Set up the appropriate equation(s)

Linear inequalities

A linear inequality is similar to a linear equation but uses inequality symbols ( $<$ , $>$ , $\leq$ , $\geq$ ) instead of an equals sign. The variable still has an exponent of 1.

Critical rule for inequalities

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When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.

Examples:

If $-2x > 6$ , dividing by $-2$ gives $x < -3$ (notice the sign flips)
If $x > -3$ and you multiply by $-1$ , you get $-x < 3$

Worked examples

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Worked Example 1: Linear equation

Solve: $5a - 7 = -2$

Solution:

$5a - 7 = -2$
$5a = -2 + 7$
$5a = 5$
$a = 1$

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Worked Example 2: Quadratic equation

Solve: $b^2 + 6b - 27 = 0$

Solution:

Looking for two numbers that multiply to $-27$ and add to $6$ .
These numbers are $9$ and $-3$ .
$(b + 9)(b - 3) = 0$
Therefore: $b = -9$ or $b = 3$

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Worked Example 3: Equation with fractions

Solve: $\frac{q}{5} = \frac{q}{2}$

Solution: Cross multiply: $2q = 5q$

$2q - 5q = 0$
$-3q = 0$
$q = 0$

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Worked Example 4: Linear inequality

Solve: $-3x + 2 > 8$

Solution:

$-3x + 2 > 8$
$-3x > 6$
$x < -2$ (inequality flips when dividing by $-3$ )

Exam tips

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Essential exam strategies:

Always check your solutions by substituting back into the original equation
When solving quadratic equations, remember they can have two solutions
For inequalities, be extra careful with negative coefficients
In word problems, define your variables clearly before setting up equations
Practice identifying whether you need one equation (one unknown) or two equations (two unknowns)

Remember!

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Key Points to Remember:

Linear equations have variables with exponent 1 and at most one solution
Quadratic equations have variables with exponent at most 2 and at most two solutions
Two unknowns require two equations to solve completely
Inequality direction changes when multiplying or dividing by negative numbers
Word problems need careful translation from words to mathematical expressions

Summary (Grade 10 NSC Matric Mathematics): Revision Notes

Summary

Types of equations

Linear equations

Quadratic equations

Systems of simultaneous equations

Algebraic solutions

Graphical solutions

Special types of equations

Literal equations

Word problems

Linear inequalities

Critical rule for inequalities

Worked examples

Exam tips

Remember!

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