43. Error Analysis Your friend used a graphing calculator to solve a system of linear equations, shown below. After using the TABLE feature, your friend says that the system has no solution. Explain what your friend did wrong. What is the solution of the system?

    
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44. Reasoning Is it possible for an inconsistent linear system to contain two lines with the same y-intercept? Explain.
45. Writing Summarize the possible relationships for the y-intercepts, slopes, and number of solutions in a system of two linear equations in two variables.

Reasoning Determine whether each statement is always, sometimes or never true for the following system.

{ y = x + 3 y = m x + b left brace . table with 2 rows and 1 column , row1 column 1 , y equals x plus 3 , row2 column 1 , y equals m x plus b , end table

46. If m = 1, the system has no solution.
47. If b = 3, the system has exactly one solution.
48. If m ≠ 1 , m not equal to 1 comma  the system has no solution.
49. If m ≠ 1 m not equal to 1  and b = 2, the system has infinitely many solutions.

C Challenge

Open-Ended Write a second equation for each system so that the system will have the indicated number of solutions.

50. infinite number of solutions

    { x 4 + y 3 = 1 ? _ left brace . table with 2 rows and 1 column , row1 column 1 , x over 4 , plus , y over 3 , equals 1 , row2 column 1 , modified question mark with under bar below , end table

51. no solutions

    { 5 x + 2 y = 10 ? _ left brace . table with 2 rows and 1 column , row1 column 1 , 5 x plus 2 y equals 10 , row2 column 1 , modified question mark with under bar below , end table

52. Write a system of linear equations with the solution set { ( x , y ) | y = 5 x + 2 } . left brace open x comma y close vertical line y equals 5 x plus 2 right brace .
53. Reasoning What relationship exists between the equations in a dependent system?
54. Economics Research shows that in a certain market only 2000 widgets can be sold at $8 each, but if the price is reduced to $3, then 10,000 can be sold.
    1. Let p represent price and n represent the number of widgets. Identify the independent and dependent variables.
    2. Write a linear equation that relates price and the quantity demanded. This type of equation is called a demand equation.
    3. A shop can make 2000 widgets for $5 each and 20,000 widgets for $2 each. Use this information to write a linear equation that relates price and the quantity supplied. This type of equation is called a supply equation.
    4. Find the equilibrium point where supply is equal to demand. Explain the meaning of the coordinates of this point within the context of the exercise.

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