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)
x
2
3
x = 2*3
x=6
(vi)
x x
 2
3 2
2 x  3x
2
6
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
1
(iv)
1
y  2x  4
5
y = 10x+20
-10x = 20-y
10x = y-20
x = y/10-2
(v)
x 2

3 3
x2
y
3
y
3y = x+2
-x = 2-3y
x = 3y-2
(vi)
2
1
y  x2
3
6
2 y x  12

3
6
3 x  36
2y 
6
12 y  3 x  36
-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
2
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
3
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
4
-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
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
0
1
2
3
4
5
6
7
8
=
a
2
2
2
2
2
2
2
2
2
+
b
1
1
1
1
1
1
1
1
1
*
x
-2
-1
0
1
2
3
4
5
6
9
8
7
6
5
4
3
2
1
0
-2
-1
(b) y = 3-4x
y
11
7
3
-1
-5
-9
-13
-17
-21
0
=
a
3
3
3
3
3
3
3
3
3
1
2
+
3
b
-4
-4
-4
-4
-4
-4
-4
-4
-4
4
5
*
6
x
-2
-1
0
1
2
3
4
5
6
6
15
10
5
0
-2
-1
-5
0
1
2
3
4
5
6
-10
-15
-20
-25
(c) y = 0.5x-2
y
=
-3
-2.5
-2
-1.5
-0
-0.5
0
0.5
1
a
-2
-2
-2
-2
-2
-2
-2
-2
-2
+
b
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
*
x
-2
-1
0
1
2
3
4
5
6
1.5
1
0.5
0
-2
-1 -0.5 0
1
2
3
4
5
6
-1
-1.5
-2
-2.5
-3
-3.5
7
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
0
-1
0
2
-2
-3
-4
-5
-6
(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
8
6
5
4
3
2
1
0
0
(c)
10
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
20
15
10
5
0
0
3
9