If you have a table that shows a linear relationship when can you read the value for b, in $y = mx + b$, directly from the table without drawing a graph or doing any calculations? Why?

Hint: The equation of a straight line in slope-intercept form is: $y = mx + b$. Where m is the value of slope and b is the y-intercept. Here, m and b are constants, and x and y are variables. If we have a table that shows a linear relationship (x, y) we can read the value for b. A table has a bunch of x values and their corresponding y values. When we take the value of x as 0, we will get the value of b.

Complete step-by-step answer:
Let us take a table that gives us a bunch of x values and their corresponding y values.
To be a linear relationship, the points have to match up in a straight line equation$y = mx + b$relationship.
Let us take the value of x as equal to 0.
Substitute the value x is equal to 0 in a straight line equation$y = mx + b$.
$ \Rightarrow y = mx + b$
Substitute the value of x is equal to 0.
$ \Rightarrow y = m\left( 0 \right) + b$
Let us simplify the above equation.
$ \Rightarrow y = b$
Hence, we can say that the value of b is equal to the value of y coordinate.

So, we can find the value of b if the value of the x coordinate is 0. Because when we put the value of the x is equal to 0, then the term ‘mx’ will become 0.

Note:
Linear relationship: The formal term to describe a straight-line graph is linear, whether or not it goes through the origin, and the relationship between the two variables is called a linear relationship. Similarly, the relationship shown by a curved graph is called non-linear.