Find the solution to each system. {2x - 3y = 4 3x - 4y = 5 {3x - 4y = 5 4x - 5y

Question

Find the solution to each system. {2x - 3y = 4 3x - 4y = 5 {3x - 4y = 5 4x - 5y = 6 {4x - 5y = 5 5x - 6y = 7 What do you notice about the coefficients of the variables and the constants on each system? What do you predict the solution for the system 5x - 6y = 7 and 6x - 7y = 8 will be? Explain your answer. Write the equations in Exercise 1 in slope-intercept form. How do the slopes compare? What is happening to the slopes of the lines in Exercises 1-3? Toward what number are the slopes approaching? What would the graph of the six equations in Exercises 1-3 look like? Find the solution to the system in Exercise 9. Then, write two systems for Exercise 10 and 11 that have the same solution. Verify your equations by solving those systems. {2x + 3y = 4 3x + 4y = 5

Explanation / Answer

1)
2x - 3y = 4   ....eqn 1
3x - 4y = 5   ....eqn 2
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multiply eqn 1 by 3 and eqn 2 by 2
6x - 9y = 12
6x - 8y = 10

subtract them
-y = 12-10
y = -2

put this in eqn 1
2x - 3y = 4
2x - 3*(-2) = 4
2x + 6 = 4
2x = -2
x = -1
Answer:
x = -1
y = -2

2)
3x - 4y = 5    ....eqn 1
4x - 5y = 6 ..eqn 2
---------------------------------
multiply eqn 1 by 4 and eqn 2 by 3
12x - 16y = 20
12x - 15y = 18

subtract them,
-y = 20-18
y = -2

put this in eqn 1:
3x - 4y = 5
3x - 4*(-2) = 5
3x + 8 = 5
3x = -3
x = -1

Answer:
x= - 1
y = -2

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