Three students are working to find the solution set of this system of equations:
3y = 6x + 6
y = 2x + 2
Use the drop-down menus to complete the statements about each of their methods.
Pedro Dropdown 1: Intersect, Do not Intersect, Are the same line.
Pedro Dropdown 2: One, Zero, Infinitely many
Amy Dropdown: is, is not
Matt Dropdown 1: Sometimes, Never, Always
Matt Dropdown 2: One, Zero, Infinitely Many
The Correct Answer and Explanation is:
To solve the system of equations and complete the statements about each student’s method, we begin by analyzing the given equations:
- ( 3y = 6x + 6 )
- ( y = 2x + 2 )
We can rewrite the first equation in slope-intercept form to facilitate easier comparison with the second equation.
Step 1: Rewriting Equation 1 in Slope-Intercept Form
The first equation is:
[
3y = 6x + 6
]
Divide both sides by 3:
[
y = 2x + 2
]
Now, both equations are:
[
y = 2x + 2 \quad \text{and} \quad y = 2x + 2
]
Step 2: Analyzing the System
Both equations represent the same line, as they have identical slopes and y-intercepts. This means the two lines overlap completely.
Step 3: Answering Pedro’s Statements
Pedro’s Statement 1:
Pedro needs to decide whether the two lines intersect or are the same. Since both equations represent the same line, they do not intersect at just one point but rather overlap entirely. Therefore, the correct choice for Pedro’s first statement is:
Pedro Dropdown 1: Are the same line
Pedro’s Statement 2:
Since the lines are the same, they intersect infinitely. Therefore, the correct choice for Pedro’s second statement is:
Pedro Dropdown 2: Infinitely many
Step 4: Answering Amy’s Statement
Amy needs to determine whether the system is or is not consistent. A system of equations is consistent if there is at least one solution. Since both equations represent the same line, there are infinitely many solutions. Therefore, the correct choice for Amy is:
Amy Dropdown: is
Step 5: Answering Matt’s Statements
Matt is dealing with the general case, where the two lines might sometimes intersect, never intersect, or always be the same.
Matt’s Statement 1:
Since the lines are the same in this case, they will always coincide, meaning the system will always have solutions. Therefore, the correct choice for Matt’s first statement is:
Matt Dropdown 1: Always
Matt’s Statement 2:
Since the two lines overlap, there are infinitely many solutions. Therefore, the correct choice for Matt’s second statement is:
Matt Dropdown 2: Infinitely many
Final Answers:
- Pedro Dropdown 1: Are the same line
- Pedro Dropdown 2: Infinitely many
- Amy Dropdown: is
- Matt Dropdown 1: Always
- Matt Dropdown 2: Infinitely many
Explanation:
The system of equations represents two lines that have the same slope (2) and the same y-intercept (2). When two lines in a system of linear equations have identical slopes and intercepts, they coincide, meaning they are the same line. Therefore, the system has infinitely many solutions, as every point on the line is a solution to both equations. This is why Pedro, Amy, and Matt all conclude that the lines are the same, the system is consistent, and there are infinitely many solutions.