FHSST: Grade 10 Math by Ewald Zietsman - HTML preview

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Chapter 10. Equations and inequalities

Strategy for Solving Equations

This chapter is all about solving different types of equations for one or two variables. In general, we want to get the unknown variable alone on the left hand side of the equation with all the constants on the right hand side of the equation. For example, in the equation x–1=0, we want to be able to write the equation as x=1.

As we saw in rearranging equations, an equation is like a set of weighing scales that must always be balanced. When we solve equations, we need to keep in mind that what is done to one side must be done to the other.

Method: Rearranging Equations

You can add, subtract, multiply or divide both sides of an equation by any number you want, as long as you always do it to both sides.

For example, in the equation x+5–1=–6, we want to get x alone on the left hand side of the equation. This means we need to subtract 5 and add 1 on the left hand side. However, because we need to keep the equation balanced, we also need to subtract 5 and add 1 on the right hand side.

(10.1)
_autogen-svg2png-0005.png

In another example, _autogen-svg2png-0006.png, we must divide by 2 and multiply by 3 on the left hand side in order to get x alone. However, in order to keep the equation balanced, we must also divide by 2 and multiply by 3 on the right hand side.

(10.2)
_autogen-svg2png-0008.png

These are the basic rules to apply when simplifying any equation. In most cases, these rules have to be applied more than once, before we have the unknown variable on the left hand side of the equation.

Tip

The following must also be kept in mind:

  1. Division by 0 is undefined.

  2. If _autogen-svg2png-0009.png, then x=0 and y≠0, because division by 0 is undefined.

We are now ready to solve some equations!

Investigation : 4 = 3 ??

In the following, identify what is wrong.

(10.3)
_autogen-svg2png-0012.png

Solving Linear Equations

The simplest equation to solve is a linear equation. A linear equation is an equation where the power of the variable(letter, e.g. x) is 1(one). The following are examples of linear equations.

(10.4)
_autogen-svg2png-0014.png

In this section, we will learn how to find the value of the variable that makes both sides of the linear equation true. For example, what value of x makes both sides of the very simple equation, x+1=1 true.

Since the definition of a linear equation is that if the variable has a highest power of one (1), there is at most one solution or root for the equation.

This section relies on all the methods we have already discussed: multiplying out expressions, grouping terms and factorisation. Make sure that you are comfortable with these methods, before trying out the work in the rest of this chapter.

(10.5)
_autogen-svg2png-0017.png

Now we see that 2x=–1. This means if we divide both sides by 2, we will get:

(10.6)
_autogen-svg2png-0019.png

If we substitute _autogen-svg2png-0020.png, back into the original equation, we get:

(10.7)
_autogen-svg2png-0021.png

That is all that there is to solving linear equations.

Solving Equations

When you have found the solution to an equation, substitute the solution into the original equation, to check your answer.

Method: Solving Linear Equations

The general steps to solve linear equations are:

  1. Expand brackets: Expand (Remove) all brackets that are in the equation.

  2. Rearrange: "Move" all terms with the variable to the left hand side of the equation, and all constant terms (the numbers) to the right hand side of the equals sign. Bearing in mind that the sign of the terms will change from (+) to (–) or vice versa, as they "cross over" the equals sign.

  3. Group like terms: Group all like terms together and simplify as much as possible.

  4. Factorise: If necessary factorise.

  5. Write solution: Find the solution and write down the answer(s).

  6. Check: Substitute solution into original equation to check answer.

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Figure 10.1. Khan academy video on equations - 1
Exercise 1.: Solving Linear Equations

Solve for x: 4–x=4

  1. Determine what is given and what is required :

    We are given 4–x=4 and are required to solve for x.

  2. Determine how to approach the problem :

    Since there are no brackets, we can start with rearranging and then grouping like terms.

  3. Solve the problem :

    (10.8)
    _autogen-svg2png-0028.png

  4. Check the answer :

    Substitute solution into original equation:

    (10.9)
    _autogen-svg2png-0029.png

    Since both sides are equal, the answer is correct.

  5. Write the final answer :

    The solution of 4–x=4 is x=0.

Exercise 2.: Solving Linear Equations

Solve for x: 4(2x–9)–4x=4–6x

  1. Determine what is given and what is required :

    We are given 4(2x–9)–4x=4–6x and are required to solve for x.

  2. Determine how to approach the problem :

    We start with expanding the brackets, then rearranging, then grouping like terms and then simplifying.

  3. Solve the problem :

    (10.10)
    _autogen-svg2png-0036.png

  4. Check the answer :

    Substitute solution into original equation:

    (10.11)
    _autogen-svg2png-0037.png

    Since both sides are equal to –20, the answer is correct.

  5. Write the final answer :

    The solution of 4(2x–9)–4x=4–6x is x=4.

Exercise 3.: Solving Linear Equations

Solve for x: _autogen-svg2png-0042.png

  1. Determine what is given and what is required :

    We are given _autogen-svg2png-0043.png and are required to solve for x.

  2. Determine how to approach the problem :

    Since there is a denominator of (3x+1), we can start by multiplying both sides of the equation by (3x+1). But because division by 0 is not permissible, there is a restriction on a value for x. (_autogen-svg2png-0047.png)

  3. Solve the problem :

    (10.12)
    _autogen-svg2png-0048.png

  4. Check the answer :

    Substitute solution into original equation:

    (10.13)
    _autogen-svg2png-0049.png

    Since both sides are equal to 2, the answer is correct.

  5. Write the final answer :

    The solution of _autogen-svg2png-0050.png is x=0.

Exercise 4.: Solving Linear Equations

Solve for x: _autogen-svg2png-0053.png

  1. Determine what is given and what is required :

    We are given _autogen-svg2png-0054.png and are required to solve for x.

  2. Determine how to approach the problem :

    We start with multiplying each of the terms in the equation by 3, then grouping like terms and then simplifying.

  3. Solve the problem :

    (10.14)
    _autogen-svg2png-0056.png

  4. Check the answer :

    Substitute solution into original equation:

    (10.15)
    _autogen-svg2png-0057.png

    Since both sides are equal to _autogen-svg2png-0058.png, the answer is correct.

  5. Write the final answer :

    The solution of _autogen-svg2png-0059.png is,   _autogen-svg2png-0060.png.

Solving Linear Equations

  1. Solve for y: 2y–3=7Click here for the solution

  2. Solve for y: –3y=0Click here for the solution

  3. Solve for y: 4y=16Click here for the solution

  4. Solve for y: 12y+0=144Click here for the solution

  5. Solve for y: 7+5y=62Click here for the solution

  6. Solve for x: _autogen-svg2png-0072.pngClick here for the solution

  7. Solve for x: 5x=3x+45Click here for the solution

  8. Solve for x: 23x–12=6+2xClick here for the solution

  9. Solve for x: 12–6x+34x=2x–24–64Click here for the solution

  10. Solve for x: 6x+3x=4–5(2x–3)Click here for the solution

  11. Solve for p: 18–2p=p+9Click here for the solution

  12. Solve for p: _autogen-svg2png-0084.pngClick here for the solution

  13. Solve for p: _autogen-svg2png-0086.pngClick here for the solution

  14. Solve for p: –(–16–p)=13p–1Click here for the solution

  15. Solve for p: 6p–2+2p=–2+4p+8Click here for the solution

  16. Solve for f: 3f–10=10Click here for the solution

  17. Solve for f: 3f+16=4f–10Click here for the solution

  18. Solve for f: 10f+5+0=–2f+–3f+80Click here for the solution

  19. Solve for f: 8(f–4)=5(f–4)Click here for the solution

  20. Solve for f: 6=6(f+7)+5fClick here for the solution

Solving Quadratic Equations

A quadratic equation is an equation where the power of the variable is at most 2. The following are examples of quadratic equations.

(10.16)
_autogen-svg2png-0101.png

Quadratic equations differ from linear equations by the fact that a linear equation only has one solution, while a quadratic equation has at most two solutions. However, there are some special situations when a quadratic equation only has one solution.

We solve quadratic equations by factorisation, that is writing the quadratic as a product of two expressions in brackets. For example, we know that:

(10.17) ( x + 1 ) ( 2 x – 3 ) = 2 x2 – x – 3 .

In order to solve:

(10.18) 2 x2 – x – 3 = 0

we need to be able to write 2x2–x–3 as (x+1)(2x–3), w

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