Jessie graphed one of the lines in a system of equations: $y=3 x-2$. If the system has an infinite number of solutions, which statements are true? Check all that apply.
A. Any point in the coordinate plane is a solution because it has an infinite number of solutions.
B. Point $(1,1)$ is a solution because it is one of the points on the line already graphed.
Answer (1)
The system has infinite solutions, meaning both equations represent the same line: y = 3 x − 2 .
Check if the point ( 1 , 1 ) is on the line by substituting x = 1 into the equation.
Calculate y = 3 ( 1 ) − 2 = 1 , confirming that ( 1 , 1 ) is a solution.
Conclude that only points on the line are solutions, not every point in the coordinate plane, so the final answer is: Point ( 1 , 1 ) is a solution because it is one of the points on the line already graphed.
Explanation
Understanding the Problem We are given a line y = 3 x − 2 and told it's part of a system of equations with infinitely many solutions. This means the other equation in the system represents the exact same line.
Infinite Solutions For a system to have infinite solutions, both equations must represent the same line. Therefore, any point on the line y = 3 x − 2 is a solution to the system.
Checking Point (1,1) Let's check if the point ( 1 , 1 ) lies on the line y = 3 x − 2 . Substitute x = 1 into the equation: y = 3 ( 1 ) − 2 = 3 − 2 = 1 . Since the calculated y value is 1, the point ( 1 , 1 ) does indeed lie on the line.
Analyzing the Statements The statement 'Any point in the coordinate plane is a solution because it has an infinite number of solutions' is incorrect. Only points on the line y = 3 x − 2 are solutions.
Final Answer Therefore, the correct statement is: Point ( 1 , 1 ) is a solution because it is one of the points on the line already graphed.
Examples
Imagine you're trying to meet a friend, and you both have the same set of instructions to get to the meeting spot. If the instructions are identical (like having the same equation for a line), you'll meet at infinitely many points along that path. In math, this is like having a system of equations with infinite solutions, where both equations describe the same line. This concept is useful in fields like engineering, where multiple systems need to behave identically, or in economics, where different models might predict the same outcome under certain conditions.
