In this chapter you will learn to find numbers that make certain statements true. A statement about an unknown number is called an equation. When we work to find out which number will make the equation true, we say we solve the equation. The number that makes the equation true is called the solution of the equation.

Setting up equations

An equation is a mathematical sentence that is true for some numbers but false for other numbers. The following are examples of equations:

\[x + 3 = 11 \text{ and } 2^x = 8\]

\(x + 3 = 11\) is true if \(x =8\), but false if \(x =3\).

When we look for a number or numbers that make an equation true we say that we are solving the equation. For example , \(x = 4\) is the solution of \(2x + 8\) because it makes \(2x +8\)true. (Check: \(2 \times 4 = 8\)

Looking for numbers to make statements true

  1. Are the following statements true or false? Justify your answer.
    1. \(x - 3 = 0\), if \(x = -3\)
    2. \(x^3 = 8\), if \(x = -2\)
    3. \(3x = -6\), if \(x = -3\)
    4. \(3x = 1\), if \(x = 1\)
    5. \(6x + 5 = 47\), if \(x = 7\)
  2. Find the original number. Show your reasoning.
    1. A number multiplied by 10 is 80.
    2. Add 83 to a number and the answer is 100.
    3. Divide a number by 5 and the answer is 4.
    4. Multiply a number by 4 and the answer is 20.
    5. Twice a number is 100.
    6. A number raised to the power 5 is 32.
    7. A number raised to the power 4 is -81.
    8. Fifteen times a number is 90.
    9. 93 added to a number is -3.
    10. Half a number is 15.
  3. Write the equations below in words using "a number" in place of the letter symbol \(x\). Then write what you think "the number" is in each case.

    Example: \(4 +x = 23\). Four plus a number equals twenty-three. The number is 19.

    1. \(8x = 72\)

    2. \( \frac{2x}{5} = 2 \)

    3. \( 2x+ 5 = 21\)

    4. \( 12 + 9x = 30\)

    5. \(30 - 2x = 40\)

    6. \(5x + 4 = 3x+ 10\)

Solving equations by inspection

The answer is in plain sight

  1. Seven equations are given below the table. Use the table to find out for which of the given values of \(x\) it will be true that the left-hand side of the equation is equal to the right-hand side.

    You can read the solutions of an equation from a table.

    \(x\)

    -3

    -2

    -1

    0

    1

    2

    3

    4

    \(2x + 3\)

    -3

    -1

    1

    3

    5

    7

    9

    11

    \(x + 4\)

    1

    2

    3

    4

    5

    6

    7

    8

    \(9 -x\)

    12

    11

    10

    9

    8

    7

    6

    5

    \(3x -2\)

    -11

    -8

    -5

    -2

    1

    4

    7

    10

    \(10x -7\)

    -37

    -27

    -17

    -7

    3

    13

    23

    33

    \(5x +3\)

    -12

    -7

    -2

    3

    8

    13

    18

    23

    \(10 - 3x\)

    19

    16

    13

    10

    7

    4

    1

    -2
    1. \(2x + 3 = 5x + 3\)
    2. \(5x + 3 = 9 - x\)
    3. \(2x + 3 = x+4\)
    4. \(10x -7 = 5x + 3\)
    5. \(3x -2 = x+4\)
    6. \(9 -x = 2x + 3\)
    7. \(10 -3x = 3x - 2\)

Two or more equations can have the same solution. For example, \(5x = 10\) and \(x + 2 = 4\) have the same solution; \(x = 2\) is the solution for both equations.

Two equations are called equivalent if they have the same solution.

  1. Which of the equations in question 1 have the same solutions? Explain.
  2. Complete the table below. Then answer the questions that follow.

    You can also do a search by narrowing down the possible solution to an equation.

    \(x\)

    0

    5

    10

    15

    20

    25

    30

    35

    40

    \(2x + 3\)

    \(3x -10\)
    1. Can you find a solution for \(2x + 3 = 3x - 10\) in the table?
    2. What happens to the values of \(2x + 3\) and \(3x - 10\) as \(x\) increases? Do they become bigger or smaller?
    3. Is there a point where the value of \(3x - 10\) becomes bigger or smaller than the value of \(2x + 3\) as the value of \(x\) increases? If so, between which \(x\)-values does this happen?

      This point where the two expressions are equal is called the break-even point.

    4. Now that you narrowed down where the possible solution can be, try other possible values for \(x\) until you find out for what value of \(x\) the statement \(2x + 3 = 3x - 10\) is true.

      "Searching" for the solution of an equation by using tables or by narrowing down to the possible solution is called solution by inspection.

More examples

Looking for and checking solutions

  1. What is the solution for the equations below?
    1. \(x - 3 = 4\)
    2. \(x + 2 = 9\)
    3. \(3x = 21\)
    4. \(3x + 1 = 22\)

When a certain number is the solution of an equation we say that the number satisfies the equation. For example, \(x=4\) satisfies the equation \(3x=12\) because \(3 \times 4 = 12\).

  1. Choose the number in brackets that satisfies the equation. Explain your choice.
    1. \(12x = 84\) {5; 7; 10; 12}

    2. \( \frac{84}{x} = 12\) {-7; 0; 7; 10}

    3. \(48 = 8k + 8\) {-5; 0; 5; 10}

    4. \(19 - 8m = 3\) {-2; -1; 0; 1; 2}

    5. \(20 = 6y - 4\) {3; 4; 5; 6}

    6. \(x^3 = -64\) {-8; -4; 4; 8}

    7. \(5^x =125\) {-3; -1; 1; 3}

    8. \(2^x = 8\) {1; 2; 3; 4}

    9. \(x^2 = 9\) {1; 2; 3; 4}

  2. What makes the following equations true? Check your answers.
    1. \(m + 8 = 100\)
    2. \( 80 = x + 60 \)
    3. \(26 - k = 0 \)
    4. \(105 \times y = 0\)
    5. \( k \times 10 = 10 \)
    6. \(5x = 100\)
    7. \( \frac{15}{t} = 5 \)
    8. \( 3 = \frac{t}{5} \)
  3. Solve the equations below by inspection. Check your answers.
    1. \( 12x + 14 = 50 \)
    2. \(100 = 15m + 25\)
    3. \( \frac{100}{x} =20\)
    4. \( 7m + 5 = 40\)
    5. \(2x + 8 = 10\)
    6. \(3x + 10 = 31\)
    7. \(-1 + 2x = -11\)
    8. \(2 + \frac{x}{7} = 5\)
    9. \(100 = 64 + 9x\)
    10. \( \frac{2x}{6}= 4\)