English | Español

Try our Free Online Math Solver!

Online Math Solver

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Linear Equation

Recall:

• When we studied scatter plots , we used MS Excel to graph a ‘trendline’ to find the best line through
the data points.

• We see from the line above that as the weight of the car increases, the MPG decreases.
• In fact, if we look at the equation we see it is y = –1.0965x + 67.354
• The slope of the line (–1.096, the number in front of the x) represents the relationship between the
variables weight and MPG. So as the weight increases 1 unit (100 lbs) the MPG decreases by 1.0965.
• The graph of this line represents a linear relationship between weight and MPG.

Background:

The symbol Δ (read delta) represents a change in a quantity.
• So Δx represents the change in x . It is found by
Δx = end value – starting value
• So if weight (x) goes from 400lbs to 700lbs, the change in weight ( Δx ) would be
Δx = 700 − 400 = 300 lbs
• Δx does not represent multiplication !

Equation of a Line:


• If we have the equation, then we have
m = 2 and b = 4
• The letters x and y are variables, meaning they vary or change along the line. At least one of them
must be nonzero. Together they represent the ordered pairs (x, y) on the graph.
• The equation is a rule that assigns a y for any x we put into the equation.
• So, we can put x = 3 into the equation like so

• And we get out a 10. This means that (3, 10) is a point on the graph of the line. We can also write

y(3) = 10

• We can find other values from by plugging in different values of x

• Putting together these (and other) points, we can graph the line it would look like the
following…

• Notice that the first point on the graph is (0,4) and the second is (1,6), and so on.
• Usually y depends on the value of x . So x is graphed on the horizontal and y on the vertical.
• m represents the slope of the line . In our example above, the slope is 2.
• b is the y-intercept, or where the line crosses the y axis. In our example above, the y- intercept is 4.

The Slope:

• m, the slope of the line can be positive or negative , large or small.
• It is a measure of
.
• Let’s find the slope of the line between the points (1,6) and (3,10). Notice these are points on our
example line above.



• This would be true for any two points on the line. Try it for (3,10) and (7, 18)

Generating a Table and Graph from an Equation:

• Generate a table and graph for
When x = 0, . This is Line 1 on our table, and the point (0,1) on the plot
When x = 2, . This is the Line 2 on our table, and the point (2,7) on the plot.

Verify the rest of the points and then draw a line through them.
• Notice that for the above problem, the slope (m) is 3, the y-intercept (b) is 1.

Determining Linear Relationships:

• There are several ways to determine a relationship is linear
− The graph (picture) is a line
− The equation is given and it is linear in form (y = mx + b)
− You are told something increases (or decreases) by a constant or fixed amount

Prev Next