Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 09, 2023

Given the point $(7,3)$ on the graph of an equation, the statement that must be true is: $x=7$ and $y=3$ make the equation true.

STEP 1

Assumptions
1. The point given is $(7,3)$ .
2. The point lies on the graph of an equation.

STEP 2

Understand what it means for a point to be on the graph of an equation.
A point $(x, y)$ is on the graph of an equation if and only if when we substitute $x$ for the $x$ -variable in the equation and $y$ for the $y$ -variable, the equation is satisfied (i.e., both sides of the equation are equal).

STEP 3

Analyze each statement to determine which one must be true given the point $(7,3)$ is on the graph of the equation.
A. The values $x=7$ and $y=3$ make the equation true.
This statement means that when we substitute $x$ with 7 and $y$ with 3 in the equation, the equation holds true. This is consistent with the definition of a point lying on the graph of an equation.

STEP 4

Analyze statement B.
B. There are solutions to the equation for the values $x=7$ and $x=3$ .
This statement is suggesting that both $x=7$ and $x=3$ are solutions to the equation. However, we are given only one point $(7,3)$ , which corresponds to a single solution where $x=7$ and $y=3$ . The statement does not necessarily follow from the information given.

STEP 5

Analyze statement C.
C. The values $x=3$ and $y=7$ make the equation true.
This statement is incorrect because it reverses the $x$ and $y$ values of the given point. The point given is $(7,3)$ , not $(3,7)$ , so this statement does not have to be true.

STEP 6

Analyze statement D.
D. The values $x=7$ and $y=3$ are the only values that make the equation true.
This statement is not necessarily true because an equation can have multiple points that satisfy it. The given point $(7,3)$ is just one of possibly many solutions to the equation.

STEP 7

Determine which statement must be true.
Based on the analysis, the only statement that must be true given the point $(7,3)$ is on the graph of the equation is statement A.
The values $x=7$ and $y=3$ make the equation true.
Therefore, the correct answer is: A. The values $x=7$ and $y=3$ make the equation true.

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