Revision Exercises



Revision Exercise Sheet - Solutions

1. Solve for x in the following equations:

(i) 2x=1

x = 1/2

(ii) 2x+6=12

2x = 12-6

2x = 6

x = 6/2

x = 3

(iii) 3-x=7

-x = 7-3

-x = 4

x = -4

(iv) 5(2x-4)=30

10x-20 = 30

10x = 30+20

10x = 50

x = 50/10

x = 5

(v) [pic]

x = 2*3

x = 6

(vi) [pic]

[pic]

2x-3x = 2*6

-x = 12

x = -12

2. Express x in terms of y in each of the following cases:

(i) y=6x+12

-6x = 12-y

6x = y-12

x = y/6-2

(ii) 6x-12y = 18

6x = 18+12y

x = 3+2y

(iii) 2x+8y=12

2x = 12-8y

x = 6-4y

(iv) [pic]

y = 10x+20

-10x = 20-y

10x = y-20

x = y/10-2

(v) [pic]

[pic]

3y = x+2

-x = 2-3y

x = 3y-2

(vi) [pic]

[pic]

[pic]

[pic]

-3x = 36-12y

3x = 12y-36

x = 4y-12

3. Solve the following systems of simultaneous equations:

(i) y=3-2x

y=x

STEP 1: Express in terms of same value of one variable (y)

(Given in question)

STEP 2: Substitute value of eq1 for eq2

3-2x = x

STEP 3: Collect the terms

3 = x+2x

3 = 3x

1 = x

STEP 4: Compute y

Sub into equation 1 or equation 2:

y = 3-2(1)

y = 1

(ii) 5x-2y = 11

3x+3y = 15

STEP 1: Express in terms of same value of one variable (15x)

(5x-2y = 11) multiplied by 3

(3x+3y = 15) multiplied by 5

15x-6y = 33

15x = 33+6y

15x+15y = 75

15x = 75-15y

STEP 2: Substitute value of eq1 for eq2

33+6y = 75-15y

STEP 3: Collect the terms

6y+15y = 75-33

21y = 42

y = 2

STEP 4: Compute x

Sub into equation 1 or equation 2:

5x-2y = 11

5x = 11+2y

5x = 11+2(2)

5x = 11+4

5x = 15

x = 3

(iii) 2x-5y=20

2=3x-2.5y

STEP 1: Express in terms of same value of one variable (5y)

2x-5y = 20

-5y = 20-2x

5y = 2x-20

(2 = 3x-2.5y) multiplied by 2

4 = 6x-5y

5y = 6x-4

STEP 2: Substitute value of eq1 for eq2

2x-20 = 6x-4

STEP 3: Collect the terms

2x-6x = -4+20

-4x = 16

4x = -16

x = -4

STEP 4: Compute y

Sub into equation 1 or equation 2:

5y = 2x-12

5y = 2(-4)-7

5y = -8-7

5y = -15

y = -3

(iv) 4P-3Q = 5

2Q+2P = 20

STEP 1: Express in terms of same value of one variable (4P)

4P = 5+3Q

(2Q+2P = 20) multiplied by 2

4Q+4P = 40

4P = 40-4Q

STEP 2: Substitute value of eq1 for eq2

5+3Q = 40-4Q

STEP 3: Collect the terms

3Q+4Q = 40-5

7Q = 35

Q = 5

STEP 4: Compute P

Sub into equation 1 or equation 2:

4P-3Q = 5

4P = 5+3Q

4P = 5+3(5)

4P = 20

P = 5

(v) x-y+z = 0

2y-2z = 2

-x+2y+2z = 29

STEP 1: Express in terms of same value of one variable (y)

Equation 1: x-y+z = 0

-y = -x-z

y = x+z

Equation 2: (2y-2z = 2) divided by 2

y-z = 1

y = 1+z

STEP 2: Substitute value of eq1 for eq2

x+z = 1+z

STEP 3: Collect the terms

x = 1+z-z

x = 1

STEP 4: Compute 2 equations for y and z

Sub into equation 3:

-x+2y+2z = 29

-1+2y+2z = 29

2y+2z = 29+1

2y+2z = 30

Eq2 and Eq3 are now 2 equations in 2 unknowns begin procedure again

2y-2z = 2

2y+2z = 30

STEP 1: Express in terms of same value of one variable (2y)

Equation 1: 2y = 2+2z

Equation 2: 2y = 30-2z

STEP 2: Substitute value of eq1 for eq2

2+2z = 30-2z

STEP 3: Collect the terms

2z+2z = 30-2

4z = 28

z = 7

STEP 4: Compute y

Sub into equation 1:

2y = 2+2z

2y = 2+2(7)

2y = 2+14

2y = 16

y = 8

(vi) 2x+2y-5z = -5

x-y+z = 3

-3x+y+2z = -2

STEP 1: Express in terms of same value of one variable (2x)

Equation 1: 2x+2y-5z = -5

2x = -5-2y+5z

Equation 2: (x-y+z = 3) multiplied by 2

2x-2y+2z = 6

2x = 6+2y-2z

STEP 2: Substitute value of eq1 for eq2

-5-2y+5z = 6+2y-2z

STEP 3: Collect the terms

-2y+5z-2y+2z = 6+5

-4y+7z = 11

1 equation in 2 unknowns. Repeat procedure for 2 other equations

STEP 1: Express in terms of same value of one variable (-3x)

Equation 2: (x-y+z = 3) multiplied by –3

-3x+3y-3z = -9

-3x = -9-3y+3z

Equation 3: -3x+y+2z = -2

-3x = -2-y-2z

STEP 2: Substitute value of eq1 for eq2

-9-3y+3z = -2-y-2z

STEP 3: Collect the terms

-3y+3z+y+2z = -2+9

-2y+5z = 7

2 equations in 2 unknowns begin procedure again using 2 new equations

-4y+7z = 11

-2y+5z = 7

STEP 1: Express in terms of same value of one variable (-4y)

Equation 1: -4y+7z = 11

-4y = 11-7z

Equation 2: (-2y+5z = 7) multiplied by 2

-4y+10z = 14

-4y = 14-10z

STEP 2: Substitute value of eq1 for eq2

11-7z = 14-10z

STEP 3: Collect the terms

-7z+10z = 14-11

3z = 3

z = 1

STEP 4: Compute y

-4y = 11-7z

-4y = 11-7(3)

-4y = 11-21

-4y = -10

4y = 10

y = 10/4 = 2½

Compute x from original equations:

2x+2y-5z = -5

2x+2(2.5)-5(3) = -5

2x +5-15 = -5

2x-10 = -5

2x = -5+10

2x = 5

x = 5/2 = 21/5

4. Given the equations of the lines:

(a) y = 2+x

(b) y = 3-4x

(c) y = 0.5x-2

Plot each line over the interval x = -2 to x = 6

(a) y = 2+x

|y |= |a |+ |b |* |x |

|0 | |2 | |1 | |-2 |

|1 | |2 | |1 | |-1 |

|2 | |2 | |1 | |0 |

|3 | |2 | |1 | |1 |

|4 | |2 | |1 | |2 |

|5 | |2 | |1 | |3 |

|6 | |2 | |1 | |4 |

|7 | |2 | |1 | |5 |

|8 | |2 | |1 | |6 |

(b) y = 3-4x

|y |= |a |+ |b |* |x |

|11 | |3 | |-4 | |-2 |

|7 | |3 | |-4 | |-1 |

|3 | |3 | |-4 | |0 |

|-1 | |3 | |-4 | |1 |

|-5 | |3 | |-4 | |2 |

|-9 | |3 | |-4 | |3 |

|-13 | |3 | |-4 | |4 |

|-17 | |3 | |-4 | |5 |

|-21 | |3 | |-4 | |6 |

(c) y = 0.5x-2

|y |= |a |+ |b |* |x |

|-3 | |-2 | |0.5 | |-2 |

|-2.5 | |-2 | |0.5 | |-1 |

|-2 | |-2 | |0.5 | |0 |

|-1.5 | |-2 | |0.5 | |1 |

|-0 | |-2 | |0.5 | |2 |

|-0.5 | |-2 | |0.5 | |3 |

|0 | |-2 | |0.5 | |4 |

|0.5 | |-2 | |0.5 | |5 |

|1 | |-2 | |0.5 | |6 |

5. Given the equations of the following lines:

(a) 2y-5x+10 = 0

(b) x = 10-2y

(c) y+5x=15

(i) Write each of the equations in the form y = f(x)

(ii) Write down the slope and the intercept of each line

iii) Write down the inverse of each function (i.e. x = g(y))

(iv) Sketch the graph of each line

(a) 2y-5x+10 = 0

(i) 2y = 5x-10

y = 5x/2-5

ii) slope = 5/2

intercept = -5

iii) y = 5x/2-5

y+5 = 5x/2

2y+10 = 5x

2y/5+2 = x

x = 2y/5+2

iv) y = 5x/2-5

Let x = 0

y = -5

x = 2y/5+2

Let y = 0

x = 2

[pic]

(b) x = 10-2y

(i) 2y = 10-x

y = 5-x/2

ii) slope = -1/2

intercept = 5

iii) inverse function: x = 10-2y

iv) y = 5-x/2

Let x = 0

y = 5

x = 10-2y

Let y = 0

x = 10

(c) y+5x=15

i) y = 15-5x

ii) Slope = -5

Intercept = 15

iii) inverse function:

5x = 15-y

x = 3-y/5

iv) y = 15-5x

Let x = 0

y = 15

x = 3-y/5

Let y = 0

x = 3

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