Answer:
To determine which equation represents the relationship between x and y based on the given table, we can examine the pattern in the values.
Looking at the x and y pairs:
X Y
1 5
2 7
3 9
4 11
We can observe that as x increases by 1, y increases by 2. This pattern suggests a linear relationship between x and y, with a constant rate of change.
Now let's evaluate the options:
A. y = 2x + 3
If we substitute x = 1 into this equation, we get y = 2(1) + 3 = 5, which matches the first pair (1, 5). Let's check the other pairs:
For x = 2, y = 2(2) + 3 = 7, matches the second pair (2, 7).
For x = 3, y = 2(3) + 3 = 9, matches the third pair (3, 9).
For x = 4, y = 2(4) + 3 = 11, matches the fourth pair (4, 11).
B. y = 3x - 2
If we substitute x = 1 into this equation, we get y = 3(1) - 2 = 1, which does not match the first pair (1, 5). Therefore, this equation does not represent the relationship.
C. y = 4x - 1
If we substitute x = 1 into this equation, we get y = 4(1) - 1 = 3, which does not match the first pair (1, 5). Therefore, this equation does not represent the relationship.
D. y = 5x
If we substitute x = 1 into this equation, we get y = 5(1) = 5, which matches the first pair (1, 5). Let's check the other pairs:
For x = 2, y = 5(2) = 10, which does not match the second pair (2, 7).
For x = 3, y = 5(3) = 15, which does not match the third pair (3, 9).
For x = 4, y = 5(4) = 20, which does not match the fourth pair (4, 11).
Based on the analysis, the equation that represents the relationship between x and y is:
A. y = 2x + 3
Answer:
A. y = 2x + 3
Step-by-step explanation:
The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.
The slope of a linear equation is found by dividing the change in the vertical direction by the change in the horizontal direction between any two distinct points on the line.
From the given table, we can observe that for each increase of 1 unit in the x-coordinate, the y-coordinate increases by 2 units. Therefore, the slope is:
[tex]\textsf{Slope $(m$)}=\dfrac{2}{1}=2[/tex]
So, the only answer option that has a slope of 2 is:
[tex]\Large\boxed{\boxed{y=2x+3}}[/tex]
To double-check this, we can substitute x = 4 into the equation and solve for y:
[tex]y=2(4)+3\\\\y=8+3\\\\y=11[/tex]
As the y-value obtained from the equation when x = 4 matches the corresponding y-coordinate from the table, this confirms that y = 2x + 3 represents the relationship between x and y.