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Questions and Answers
Which inequality represents the values of w for which it would be faster for Adam to walk to school?
Which inequality represents the values of w for which it would be faster for Adam to walk to school?
- w + 5 > 20 (correct)
- w + 5 < 20
- w - 5 = 20
- w - 5 < 20
It takes Adam the same amount of time to walk to school as it does to take the bus when he waits for 0 minutes.
It takes Adam the same amount of time to walk to school as it does to take the bus when he waits for 0 minutes.
False (B)
What is the average mileage per gallon of gasoline for Alan's car?
What is the average mileage per gallon of gasoline for Alan's car?
25 miles
The cost of gasoline per gallon is $______.
The cost of gasoline per gallon is $______.
How much does Alan want to reduce his weekly expenditure on gasoline?
How much does Alan want to reduce his weekly expenditure on gasoline?
Which condition indicates that two lines represented by equations are parallel?
Which condition indicates that two lines represented by equations are parallel?
If Alan drives an average of 100 miles each week, how many gallons of gasoline does he use?
If Alan drives an average of 100 miles each week, how many gallons of gasoline does he use?
The total time it takes Adam to walk to school is ______ minutes.
The total time it takes Adam to walk to school is ______ minutes.
The equations -3x + 36y = 8 and x - 2py = 6 can have a solution if p is equal to 6.
The equations -3x + 36y = 8 and x - 2py = 6 can have a solution if p is equal to 6.
What value of p indicates that the system has no solution?
What value of p indicates that the system has no solution?
Match the following variables with their meanings:
Match the following variables with their meanings:
The two equations are in standard form as _____ and ______.
The two equations are in standard form as _____ and ______.
Match the following elements from the system of equations to their descriptions:
Match the following elements from the system of equations to their descriptions:
What is the result when multiplying each side of the equation y - x = 23 - 32y by 12?
What is the result when multiplying each side of the equation y - x = 23 - 32y by 12?
If the constants in the equations are proportional, then the lines are guaranteed to be distinct.
If the constants in the equations are proportional, then the lines are guaranteed to be distinct.
What mathematical operation preserves the relationship of proportionality in the equations?
What mathematical operation preserves the relationship of proportionality in the equations?
What is the cost, in dollars per mile, for Alan to drive his car?
What is the cost, in dollars per mile, for Alan to drive his car?
Alan wants to reduce his driving by 5 miles each week.
Alan wants to reduce his driving by 5 miles each week.
If Alan's car travels 25 miles per gallon, how many miles can he drive with 1 gallon of gasoline?
If Alan's car travels 25 miles per gallon, how many miles can he drive with 1 gallon of gasoline?
The number of containers of detergent, represented by ______, must be at least twice the number of containers of fabric softener represented by s.
The number of containers of detergent, represented by ______, must be at least twice the number of containers of fabric softener represented by s.
Match the following variable with what it represents in the context of Alan's problem:
Match the following variable with what it represents in the context of Alan's problem:
What is the maximum weight of the shipment from the supplier?
What is the maximum weight of the shipment from the supplier?
Each container of fabric softener weighs more than each container of detergent.
Each container of fabric softener weighs more than each container of detergent.
The total weight of detergent and fabric softener that the supplier delivers cannot exceed ______ pounds.
The total weight of detergent and fabric softener that the supplier delivers cannot exceed ______ pounds.
What is the minimum number of trips per month that a monthly pass becomes cheaper than purchasing individual tickets at $3.50 each?
What is the minimum number of trips per month that a monthly pass becomes cheaper than purchasing individual tickets at $3.50 each?
The weight of each container of detergent is 14.7 pounds.
The weight of each container of detergent is 14.7 pounds.
Express the inequality that represents the relationship between the number of containers of detergent (D) and fabric softener (F).
Express the inequality that represents the relationship between the number of containers of detergent (D) and fabric softener (F).
The price of a monthly pass is $______.
The price of a monthly pass is $______.
Why is the inequality D < 14.7F incorrect?
Why is the inequality D < 14.7F incorrect?
Purchasing 27 individual tickets costing $3.50 each is a cheaper option than a monthly pass.
Purchasing 27 individual tickets costing $3.50 each is a cheaper option than a monthly pass.
What is the cost per individual ticket for the longest trip?
What is the cost per individual ticket for the longest trip?
Match the costs of tickets with their respective trip lengths:
Match the costs of tickets with their respective trip lengths:
When does a system of two linear equations have no solution?
When does a system of two linear equations have no solution?
The coefficients in parallel lines represented by two equations are proportional.
The coefficients in parallel lines represented by two equations are proportional.
What is the value of r in the system of equations for it to have no solution?
What is the value of r in the system of equations for it to have no solution?
A system of two equations has no solution when their lines are _____ and parallel.
A system of two equations has no solution when their lines are _____ and parallel.
In the standard form of the equation Ax + By = C, which part represents the constant?
In the standard form of the equation Ax + By = C, which part represents the constant?
Match the following elements to their significance in the system of equations:
Match the following elements to their significance in the system of equations:
If the coefficient of x in one equation is 16, the coefficient of x in the other equation must also be 16 for the lines to be parallel.
If the coefficient of x in one equation is 16, the coefficient of x in the other equation must also be 16 for the lines to be parallel.
What transformation must be made to the first equation to write it in standard form?
What transformation must be made to the first equation to write it in standard form?
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Study Notes
Lines in Standard Form
- Lines represented by equations in standard form, 𝐴𝑥 + 𝐵𝑦 = 𝐶 and 𝐷𝑥 + 𝐸𝑦 = 𝐹, are parallel if the coefficients for 𝑥 and 𝑦 in one equation are proportional to the corresponding coefficients in the other equation; this means that 𝐷/𝐴 = 𝐸/𝐵.
- Lines are distinct if the constants are not proportional, meaning 𝐶/𝐴 ≠ 𝐹/𝐷.
Example: Solving for a Variable
- The provided system of equations:
- 𝑦 - 𝑥/2 = 23 - 32𝑦 (first equation)
- 𝑥 + 3/2 = 𝑝𝑦 + 9/2 (second equation)
- To determine when the system has no solution, we must find the value of 'p' that makes the lines parallel and distinct.
- Rearranging the first equation:
- 36𝑦 - 3𝑥 = 8
- -3𝑥 + 36𝑦 = 8
- Rearranging the second equation:
- 𝑥 - 2𝑝𝑦 = 6
- The two equations in the standard form are:
- -3𝑥 + 36𝑦 = 8
- 𝑥 - 2𝑝𝑦 = 6
- For the system to have no solutions, the lines must be parallel and distinct. This means the ratio of the coefficients for 𝑥 and 𝑦 in both equations must be equal (1/-3 = -2𝑝/36) and the ratio of the constants must not be equal (6/1 ≠ 8/-3).
- Solving the proportion, we get 𝑝 = 6.
- Therefore, if the system has no solution, the value of 𝑝 is 6.
Additional Notes
- The text provides a thorough explanation of how to solve for the value of a variable in a system of equations with no solution.
- It emphasizes the relationship between the coefficients and constants of the lines in standard form and how their proportions determine if the lines are parallel and distinct.
- The steps involved in manipulating the equations and simplifying them to standard form are clearly explained.
- The importance of understanding the concept of proportional coefficients in determining parallel lines is highlighted.
- The process of solving the proportion to find the value of 'p' is demonstrated, emphasizing the connection between the coefficient ratios and the no-solution condition.
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