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Solving Linear equation by graphical and other methods

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Problems

1. How are addition and multiplication used to solve a linear equation? Demonstrate by solving "15x + 7 = 31 + 3x"

2. Translate the following into a system of equations, then solve it:A customer walks into an electronics store and buys five MP3 players and eight sets of headphones, paying $840. A second customer buys three MP3 players and four sets of headphones, and pays $480. How much does an MP3 player cost? How much does a set of headphones cost?

3. (i) What are the x-intercept and y-intercept of a linear equation?
(ii) What are their coordinates on a graph?
(iii) How can they be used to graph a line?
(iv) Demonstrate by determining the intercepts of "14x + 7y = 21".

4. Write a system of equations having
a. A unique solution.
b. An infinite number of solutions.
c. No solution.
How would each system appear graphically?

5. Explain how to apply elimination in solving a system of equations.
a. Explain how to apply substitution in solving a system of equations.
b. Demonstrate each technique in solving the system
3x + 9y = 12
5x - 4y = 3

6: Linear Equations and their Solutions
From the given polynomials, identify the polynomials of degree one.
a. 311y - 5 - 43y
b. (11y2)1/2 + 14
c. 10 + (19)1/2x
d. 2 + 15x
e. 52y4 + 7x + 2
f. (68)1y1
g. x3 + 3x - 9
h. (2x)1/2 + 4x - 8

Solve the following:
i. -2x = 3x + 4
ii. 3x/4 = 6
iii. y/6 + 1 = 9
iv. 6 = -2x/4
v. Find f(1) for f(x) = 4x3 - 3x2 - x + 2
vi. A function gives the value of C as 2 _ (22/7) _ r. Find C when r = 21 cm and r = 84 cm.

7: Ratio and Proportion
1. A lawyer bills her clients $200 per hour of service. If a client's case requires 39 hours to complete, use proportion to calculate how much the client will owe the lawyer (excluding tax).
2. A new virus is released on the internet; the administrator of a department's Local Area Network ( LAN) is given five minutes by a manager to estimate the impact. The administrator samples 15 of the PCs connected to the LAN, and finds that 9 are infected; use proportion to estimate the number of infected PCs if there are a total of 202 PCs connected to the LAN.
3. An administrator of a popular web site is told that a new server can handle 41,000 "hits" (users accessing the site) per second. The web site currently experiences a peak demand of about 105,000 hits per second; but every month, the peak demand increases by 2800 hits per second. Use a proportion equation to determine how many new servers the administrator should buy to address expected traffic for the next 24 months.

8: Graph and Analyze Linear Functions
1. How do we write the equation of a horizontal line? What would be an example?
2. How do we write the equation of a vertical line? What would be an example?
3. The points (3, 9), (5, 13), (15, 33), (34, 71), (678, 1359), and (1234, 2471) all lie on line M.
The points (3, -9), (5, -11), (15, -21), (34, -40), (678, -684), and (1234, -1240) all lie on line N.
a. Form the equations of both the lines. Show your work.
b. What are the co-ordinates of the point of intersection of lines M and N?
c. Write the co-ordinates of the intersections of lines M and N with the x-axis.
d. Write the co-ordinates of the intersection of lines M and N with the y-axis.

9: Solving System of Linear Equations
1. Solve by substitution or elimination method:
3x - 2y = 8
-12x + 8y = 32
2. Solve by substitution or elimination method:
7x - 5y = 14
-4x + y = 27
3. Solve by substitution or elimination method:
-4x + 3y = 5
12x - 9y = -15

10: Graphical Representation of Linear Equations
1. Plot the graph of the equations 3x - 8y = 5 and 4x - 2y = 11 and interpret the result.
2. Plot the graph of the equations 4x - 6y = 2 and 2x - 3y = 1 and interpret the result.
3. Plot the graph of the equations 10x - 4y = 3 and 5x - 2y = 6 and interpret the result.

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Solution describes steps in solving set of linear equations by substitution, elimination and graphical methods.

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1.How are addition and multiplication used to solve a linear equation? Demonstrate by solving "15x + 7 = 31 + 3x"

A linear equation can be solved by adding or multiplication both sides by equal terms. Objective is to get one side with variable and other side with constants. Then we can make the coefficient of variable equal to one by multiplying by a suitable value to both sides and can get the value of variable.

Let us consider the following equation:
15x + 7 = 31 + 3x
We will try to bring all variable on L.H.S. by adding -3x both sides we get
15x+7-3x=31+3x-3x
12x+7=31
We are able to remove variable part from R.H.S. Now we will try to remove constant part from L.H.S.
By adding -7 to both sides, we get
12x +7-7=31-7
12x=24
Let us try to make coefficient of x equal to one. This can be done by multiplying both sides by (1/12), we get
12x*1/12 = 24*1/12
x = 24/12 = 2

2.Translate the following into a system of equations, then solve it:A customer walks into an electronics store and buys five MP3 players and eight sets of headphones, paying $840. A second customer buys three MP3 players and four sets of headphones, and pays $480. How much does an MP3 player cost? How much does a set of headphones cost?

Let the price of MP3 player is x and price of headphone is y.
Then In first case we see customer pays $840 for 5 MP3 players and 8 sets of headphones, we get
5x+8y=840 -------(1)
In second case, customer pays $480 for 3 MP3 players and 4 sets of headphones, we get
3x+4y =480 ------(2)

We see that coefficient of y is 8 in first case and 4 in the second case, by multiplying by 2 both sides of equation 2 we get
6x+8y = 960 -----(3)

On subtracting equation (3) from (1) we get

5x+8y=840
6x+8y = 960
- - -
-x = -120
or x = 120

Put the value of x in any of the equation and get the value of y
Let us take equation 1

5x+8y=840
5*120+8y = 840
600+8y = 840
8y = 240
y = 30

So, price of MP3 player is $120 and headphone set is $30

3. (i) What are the x-intercept and y-intercept of a linear equation?

Intercept x means the point where line cuts x axis. At this point y is 0.

Similarly y intercept means the point where line cuts y axis. At this point x is 0.

(ii) What are their coordinates on a graph?
Coordinates are the points through which line passes.

(iii) How can they be used to graph a line?
If we are given 2 coordinates of the line, locate the two points. A line joining these two points will be required line.

(iv)Demonstrate by determining the intercepts of "14x + 7y = 21".

x- intercept i.e. line passes through a point (x,0),

14x +7(0) = 21

x = 1.5 (means x intercept coordinate is (1.5,0)

Similarly y intercept means line passes through (0,y)
14(0) + 7y =21
7y =21
y =3 (means y intercept coordinate is (0,3)

4.Write a system of equations having, How would each system appear graphically?
a.A unique solution.
X+Y =4
X-Y=2
The above set of equation will have only one solution

b.An infinite number of solutions.
X+Y =4
2x+2Y =8
The above set of equation will have Infinite number of ...

Solution provided by:
Lokesh Jain, MSc
Education
  • BEng (Hons) , Birla Institute of Technology and Science, India
  • MSc (Hons) , Birla Institute of Technology and Science, India
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