About Graphing Linear Equations & Proportional Relationships
What is a Linear equation?
A linear equation is defined as an equation with a degree of 1. Not only that, if such an equation is graphed it will always take the shape of a straight line.
A linear equation in one variable is a single-variable equation. Ax + B = 0 is the standard form of a linear equation in one variable where x is the variable, A is its coefficient and B is a constant in this equation.
A two-variable equation where the exponents of both the variables is 1 is called a linear equation in two variables. A linear equation in two variables written in the standard form is Ax + By = C; where x and y are variables, A and B are coefficients and C is a constant.
What is an ordered pair?
An ordered pair is made up of the x (abscissa) and y (ordinate) coordinates, with two values written in a specific order within parentheses; (abscissa, ordinate). It aids in the visual comprehension of a point on the Cartesian plane.
Example: (4, 6)
Graphing Linear equations
The graph of a two-variable linear equation is a straight line (hence the name linear). You can graph an equation by finding any two solutions \((x_1,~y_1)\) and \((x_2,~y_2)\), plotting these two points and drawing a line connecting them.
For example: Graph \(3x+2~=~y\)
Step 1: Create a table of values
To find ordered pairs, we substitute the values of x with -1, 0 and 1 to find the corresponding y values. Then we get 3 ordered pairs.
| \(x\) | \(3x+2~=~y\) | y | \((x,~y)\) |
|---|---|---|---|
-1 | \(y~=~3(-1)+2\) | -1 | (-1, 1) |
0 | \(y~=~3(0)+2\) | 2 | (0, 2) |
1 | \(y~=~3(1)+2\) | 5 | (1, 5) |
Create a graph of the Linear equation by plotting the ordered pairs
Now we plot these ordered pairs into the coordinate plane. These points are then joined by a straight line. This straight line is the graph of the given equation.
Below we can see the linear equation graph.
What is a Proportional relationship?
A proportional relationship between two variables \(x\) and \(y\) is when one variable is always some constant value multiplied by the other. The “constant of proportionality” is the name given to this constant. All proportional relationships can be represented by the equation \(y~=kx\) , where \(k\) is the proportionality constant.
Identifying a proportional relationship using a ratio table
Example: Tom pays $10 per month for a video streaming service. The table below shows the cost paid by Tom for a number of months.
Number of months, \(y\) | Total cost, \(x\) |
|---|---|
1 | 10 |
2 | 20 |
3 | 30 |
Let’s find the ratio of \(x\) for the corresponding values of \(y\)
Number of months, \(y\) | Total cost, \(x\) | Ratio of \(x\) to \(y\) |
|---|---|---|
1 | 10 | \(\frac{10}{1}~=~10\) |
2 | 20 | \(\frac{20}{2}~=~10\) |
3 | 30 | \(\frac{30}{3}~=~10\) |
As you can see, this is a proportional relationship between the number of months Tom pays for the streaming service, \(y\) , and the total cost of the service, \(x\) .
The values of \(‘x’\) are always 10 times the values of \(‘y’\) , which means 10 is the constant of proportionality. You can also model this relationship with the equation \(x~=~10y\) .
Identifying a proportional Proportional relationship using a graph
If a line is straight and passes through the origin, you can tell it represents a proportional relationship.
To calculate the constant of proportionality, find the ratio of the y-coordinate to the x-coordinate at any point on the line. The constant of proportionality is the same for all x and y coordinates for a given equation.
- For instance, the point (2,2) is on the line.
- The ratio of the y-coordinate to the x-coordinate is \(\frac{2}{2}=1 \).
So, the constant of proportionality is 1. So, you can represent this relationship with the equation \(y=x \)
Examples on Graphing Linear Equations & Proportional Relationships
Example 1: Graph the equation \(2x+2~=~y\)
Solution:
Step 1: Construct a table of values.
| \(x\) | \(y~=~2x+2\) | \(y\) | \((x,~y)\) |
|---|---|---|---|
-1 | \(y~=~2(-1)+2\) | 0 | (-1, 0) |
0 | \(y~=~2(0)+2\) | 2 | (0, 2) |
1 | \(y~=~2(1)+2\) | 4 | (1, 4) |
Step 2: Plot the ordered pairs.
Step 3: Now through these points, draw a straight line.
Example 2: Graph the equation \(2y=x+5\)
Solution:
\(2y=x+5\)
\(y=\frac{1}{2}~(x+5)\)
\(y=\frac{x}{2}+\frac{5}{2} \)
\(y=0.5x+ 2.5 \)
Step 1: Construct a table of values.
| \(x\) | \(x+2y~=~5 \) | \(y \) | \((x,~y) \) |
|---|---|---|---|
-1 | \(y~=~0.5(-1)+2.5 \) | 2 | (-1, 2) |
0 | \(y~=~0.5(0)+2.5 \) | 2.5 | (0, 2.5) |
1 | \(y~=~0.5(1)+2.5 \) | 3 | (1, 3) |
Step 2: Plot the ordered pairs.
Step 3: Now through these points, draw a straight line.
Example 3: The equation \(y~=~0.5x+12 \) represents the height of a tree after x years (in centimetres). In five years, how much does the tree grow? Justify your answer with a graph.
Solution:
Step 1: Construct a table of values.
| \(x \) | \(y~=~0.5x+12 \) | \(y \) | \((x,~y) \) |
|---|---|---|---|
1 | \(y~=~0.5(1)+12 \) | 12.5 | (1, 12.5) |
2 | \(y~=~0.5(2)+12 \) | 13 | (2, 13) |
3 | \(y~=~0.5(3)+12 \) | 13.5 | (3, 13.5) |
4 | \(y~=~0.5(4)+12 \) | 14 | (4, 14) |
Step 2: Plot the ordered pairs.
Step 3: Now through these points, draw a straight line.
We need to find the height of the tree in 5 years. In the graph, we can find the corresponding y coordinate to be 14.5 cm when x is 5 cm
Also by substituting \(x=5 \) in the equation we get,
\(y=0.5(5)+12=14.5 \) cm
Example 4: Graph the equation \(y=0.8x \)
Solution:
Step 1: Construct a table of values.
| \(x\) | \(y~=~0.8x \) | \(y \) | \((x,~y) \) |
|---|---|---|---|
-1 | \(y~=~0.8(-1) \) | -0.8 | (-1, -0.8) |
0 | \(y~=~0.8(0) \) | 0 | (0, 0) |
1 | \(y~=~0.8(1) \) | 0.8 | (1, 0.8) |
Step 2: Plot the ordered pairs.
Step 3: Now through these points, draw a straight line.
As you can see this line passes through the origin hence this is a proportional relationship, also the equation is of the form \(y=kx \) where k is the constant of proportionality, here \(k \) is 0.8.
Example 5: Solve the equation \(y=2x \)
Solution:
Step 1: Construct a table of values.
| \(x \) | \(y~=~2x \) | \(y \) | \((x,~y)\) |
|---|---|---|---|
-1 | \(y~=~2(-1) \) | -2 | (-1, -2) |
0 | \(y~=~2(0) \) | 0 | (0, 0) |
1 | \(y~=~2(1) \) | 2 | (1, 2) |
Step 2: Plot the ordered pairs.
Step 3: Now through these points, draw a straight line.
As you can see this line passes through the origin hence this is a proportional relationship, also the equation is of the form \(y~=kx \) where \(k \) is the constant of proportionality, here \(k \) is 2.
Example 6: The cost (in dollars) for ounces of frozen meat is represented by \(y~=5x \) , where y represents the cost for \(x \) ounces. Interpret the slope by graphing the equation.
Solution:
Step 1: Construct a table for the cost of the meat.
| \(x \) | \(y~=5x \) | \(y\) | \((x,~y) \) |
|---|---|---|---|
1 | \(y~=5(1) \) | 5 | (1, 5) |
2 | \(y~=5(2) \) | 10 | (2, 10) |
3 | \(y~=5(3) \) | 15 | (3, 15) |
4 | \(y~=5(4) \) | 20 | (4, 20) |
Step 2: Plot the ordered pairs.
Step 3: Now through these points, draw a straight line.
As you can see, this line passes through the origin hence this is a proportional relationship, also the equation is of the form \(y~=kx \) where \(k\) is the constant of proportionality. In this case, the value of \(k\) is 5. The constant of proportionality is also the slope hence the slope is 5 dollars per ounce.
A line’s slope is its steepness. Slope is the corresponding change in y for a specific change in x in an equation.
A solution is a value or a set of values when substituted for the unknown makes the equation true. In other words, the system of linear equation solution sets is the collection of all possible values for the variables that satisfy the given linear equation.
In a proportional relationship \(y~=kx \) , the constant \(k\) is known as the proportionality constant, the unit rate, and/or the slope. If \(x \) equals 1, then \(y \) equals \(k \) in the equation. When \(x \) equals 1 on a graph of a proportional relationship, the corresponding y-value is the slope or the unit rate of the graph.