Everything Maths Grade 10. Chapter 2: Equations and inequalities.

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4. Equations and inequalities
Grade 10

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Linear equations
● A linear equation is an equation where the highest exponent of the variable is 1:
2x + 2 = 1
4(2x – 9) – 4x = 4 – 6x
● A linear equation has at most one solution (the solution is also called the root).
● The general steps for solving linear equations are:
1. Expand all brackets.
2. Rearrange the terms so that all variable terms are on one side and all constant terms
are on the other side.
3. Group like terms together and simplify.
4. Factorise if necessary.
5. Find the solution and write down the answer.
6. Check the answer by substituting back into the original equation.

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Quadratic equations
● A quadratic equation is an equation where the exponent of the variable is at most 2:
2x2
+ 2x = 1
3x2
- 4 = 0
● A quadratic equation has at most two solutions.
● The general steps for solving quadratic equations are:
1. Rewrite the equation in the required form, ax2
+ bx + c = 0.
2. Divide the entire equation by any common factor of the coefficients to obtain an
equation of the form ax2
+ bx + c = 0.
3. Factorise ax2
+ bx + c = 0 to be of the form (rx + s)(ux + v) = 0.
4. The two solutions are (rx + s) = 0 or (ux + v) = 0, so
5. Check the answer by substituting it back into the original equation.

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Simultaneous equations
● To solve for two unknown variables, two equations are solved simultaneously:
x + y = -1 and 3 = y - 2x
● Linear simultaneous equations can be solved algebraically using substitution or
elimination methods.
● We use the two given equations to eliminate one variable and then solve.
● To solve graphically we draw the graph of each equation. The solution to the system of
equations will be the coordinates of the point of intersection.

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Word problems
● Word problems require a set of equations that represent the problem mathematically.
1. Read the whole the question
2. What are we asked to solve for?
3. Assign a variable to the unknown quantity, for example x.
4. Translate the words into algebraic expressions by rewriting the given information
in terms of the variable.
5. Set up an equation or system of equations to solve for the variable.
6. Solve the equation algebraically using substitution.
7. Check the solution.

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Literal equations
● Literal equations are equations that have several letters and variables.
● An example is the area of a circle (A = пr2
)
● To make one particular variable the subject of the formula, we rearrange the equation
so that the required variable is on its own.
1. We isolate the unknown variable by asking “what is it joined to?” and “how is it
joined?”. We then perform the opposite operation to both sides as a whole.
2. If the unknown variable is in two or more terms, then we take it out as a common
factor.
3. If we have to take the square root of both sides, remember that there will be a
positive and a negative answer.
4. If the unknown variable is in the denominator, we multiply both sides by the LCD and
then continue to solve.

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Linear inequalities
● A linear inequality is similar to a linear equation and has the exponent of the variable
equal to 1.
2x + 2 < 1
5x – 6 > 7x + 5
● If we divide or multiply both sides of an inequality by any number with a minus
sign, the direction of the inequality changes.
● Solutions to linear inequalities can be represented on a number line or in interval
notation.

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