Answer:
[tex]\displaystyle y=8x[/tex]
Step-by-step explanation:
We are given a table and we want to determine a linear equation (in slope-intercept form) that represents this table.
So, we will first need to determine the slope. We can use the slope formula:
[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}[/tex]
Where (x₁, y₁) and (x₂, y₂) are two points.
We have three points: (1, 8), (2, 16), and (3, 24).
Let’s pick two. We can use (1, 8) and (2, 16).
So, let (1, 8) be (x₁, y₁) and let (2, 16) be (x₂, y₂). Substitute appropriately:
[tex]\displaystyle m=\frac{16-8}{2-1}[/tex]
Evaluate:
[tex]\displaystyle m=\frac{8}{1}=8[/tex]
So, our slope is 8.
Now, we can determine the equation using point-slope form:
[tex]\displaystyle y-y_1=m(x-x_1)[/tex]
Where m is our slope and (x₁, y₁) is a point.
We know that the slope is 8, so we will substitute 8 for m.
We can use any of our three points for the point. Let’s use (1, 8) for consistency. So, let (1, 8) be (x₁, y₁). Substitute appropriately:
[tex]\displaystyle y-8=8(x-1)[/tex]
Solve for y. Distribute the right:
[tex]\displaystyle y-8=8x-8[/tex]
Add 8 to both sides:
[tex]\displaystyle y=8x[/tex]
And we have our equation.
Notice that this is a direct variation or proportional equation as we don’t have a y-intercept.
