Linear Equations

What are Linear Equations?

In the functions section we talk about how a function is like a box that takes an independent input value and uses a rule defined mathematically to create a unique output value. The value for the output is dependent on the value that is put in the box. We call the values that are going into the box the independent values or the domain. We call the values coming out of the box the dependent values or the range.

Unless we specify differently, on Cartesian graphs the domain is the real numbers. In the Cartesian Plane section we saw how running different values through a function to identify the points on the Cartesian plane by picking the first (x) value of the point the domain and the second (y) value from the range. To restate this: by convention the two variables for a function on the Cartesian Plane are x for the domain, the independent variable, and y for the range, the dependent variable. The variable y is the same as writing f(x). Mathematicians recognize this equivalence but generally prefer to write y because its shorter. Because a function definition has an input and an output it must also contain an equal sign. The section in this book solving equations showed the various operations we can perform on both sides of an equal sign and still maintain the notion of equivalence. In this section we plug different values into the independent variable and solve to find the associated dependent variable. For instance if we start with the equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = x} We can add a -x to both sides to get the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y -x = x - x } which we then simplify to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y -x = 0 } Or we can add a -y to both sides to get the equation $y-y=x-y$ which we then simplify to $0=x-y$ Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y -x = 0 \equiv 0 = x - y} then: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y -x = 0 = x - y} Using the transitive property Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y -x = x - y} adding x + y to both sides gives us $(y-x)+(y+x)=(x-y)+(y+x)$ using the associative property we change this to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y - y )+( x + x) = (x - x) + (y + y)} which simplifies to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x = 2y } And divide both sides by 2 $2x/2=2y/2$ To simply reverse the order and show Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = y}

We have not really proved anything mathematically above, but these operations allow us to manipulate equations to get the dependent variable by itself on one side of the equals sign. Then we can plug numbers into the independent variable to discover the function values for those numbers. Then we can draw these values as points on the Cartesian plane and get a feel for what the function would look like if we could see all the points defined by the function at once.

Horizontal Linear Equations

Equations of the form y = C₂ are linear functions of the general form y = m x + b where slope m = 0 and the constant C₂ is the y-intercept b (in the general form). The graph of this zero-slope function is a straight horizontal line, intercepting the y-axis at C₂, including zero and extends infinitely in the positive and negative directions for all R values of x (see the following diagram).

The domain for such functions is R covering all real numbers (unless otherwise specified), but the range is just the set { c }. The equation y = 0 is the x-axis.

Vertical Linear Equations

Equation x = C₁, x is one single value C₁ and y, being unrestricted, is every R number. The graph of x = C₁ is a straight vertical line where x = C₁, covering all positive, negative and zero values of y (see the following diagram).

Its domain is set { C₁ } and range is set R (unless otherwise specified). x = C₁ is technically not a function (there is more than one value of y for each value of x), but it's a relation. Vertical lines have no slope (m = divide by zero, undefined, plus and minus infinity). These are the only types of linear equations of the general form shown previously which are not linear functions. The equation x = 0 is exactly the y-axis. Lines with steepness approaching vertical have very large-magnitude slopes but are still functions.

Linear Equations

We are going to start by looking at simple functions called linear equations. When none of the instances of x and y in the algebraic expression defining the function rule have exponents then all the instances of x and y can be combined into just two occurrences. the graph of the expression can be represented as a straight line. The equation that expresses the function is considered a linear equation with two variables. The following equation is a simple example of such a linear equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - x = 2 \, }

Since y is the dependent variable it is standing in for the function. We can re-write the expression as f(x) - x = 2. If we add an x to both sides the equality property holds and we get the expression f(x) - x + x = x + 2. Simplifying we get f(x) = x + 2. In the following table we'll pick 3 values for x, and then calculate the dependent (y) values from f(x).

x value	y value (abscissa)	Coordinates (x,y)
-1	1	(-1,1)
0	2	(0,2)
1	3	(1,3)

where x and y are variables to be plotted in a two-dimensional Cartesian coordinate graph as shown here:

This function is equivalent to the previous example of a linear equation, y - x = 2. The arrows at each end of the line indicate that the line extends infinitely in both directions. All linear functions of a single input variable have or can be algebraically arranged to have the general form:

$y=f(x)=mx+b\,$

where x and y are variables, f(x) is the function of x, m is a constant called the slope of the line, and b is a constant which is the ordinate of the y-intercept (i. e. the value of y where the function line crosses the y-axis). The slope indicates the steepness of the line. In the previous example where y = x + 2, the slope m = 1 and the y-intercept ordinate b = 2. The y = mx+b form of a linear function is called the slope-intercept form.

Unless a domain for x is otherwise stated, the domain for linear functions will be assumed to be all real numbers and so the lines in graphs of all linear functions extend infinitely in both directions. Also in linear functions with all real number domains, the range of a linear function will cover the entire set of real numbers for y, unless the slope m = 0 and the function equals a constant. In such cases, the range is simply the constant.

Conversely, if a function has the general form y = mx + b or if it can be arranged to have that form, the function is linear. A linear equation with two variables has or can be algebraically rearranged to have the general form¹:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A x + B y = C \, }

where x and y represent the linear variables, and the letters A, B, and C can represent any real constants, either positive or negative. Conversely, an equation with two variables x and y having that general form, or being able to be arranged in that form, would be linear as long as A and B are not both equal to 0. In the preceding equation, capital letters are to avoid confusion with other constants in this chapter and for consistency with Reference 1.

If one divides the preceding equation by B (when B is not 0) and solves for y, the following form can be obtained:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = (-A/B) x + (C/B) \, }

If one equates -A/B to the slope m and C/B to the y-intercept ordinate b, it can be seen that the general form for a linear equation and the slope-intercept form for a linear function are practically interconvertible except for the fact that, in a linear function, the B constant in the linear equation form cannot equal 0.

X and Y axis intercepts

An axis intercept point is a point where the graph of a function, relation, or equation intersects the X or Y axes. This section is about finding out how a particular set of functions: linear functions cross the axes.

We know that the domains of most lines are infinite because they are defined at every value of X. The exception is lines that are defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=c} where c is a number we choose when we write the function. By definition this line is only defined for one value of X. Since the domain maps onto more than one value for the range this is actually a relationship and not a function. We've seen the graph of this relationships is a vertical line that passes through the point (c,Y).

Lines with the equation X=C intersect the X axis once and the Y axis 0 or infinite times.

We can also restrict the range of a function by simply writing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y=c} where c is again any number we choose. The graph of this line is a horizontal line that passes through the point (X,c). When Y=o this line is the same as the X axis. when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \ne 0 } , then the line can never intercept the X axis.

Lines with the equation Y=C intersect the Y axis once and the X axis 0 or infinite times.

When looking at Cartesian graphs and linear equations we run into a mathematical axiom: "Two points determine a line.". We will see how this axiom affects the slope-intercept definition of a line $y=f(x)=mx+b$ in the next section. When two lines intersect they intersect at a point. If a line is not horizontal or perpendicular it will have to intersect the X and Y axes once, but only once.

In this page we are going to accept the statement"At most one line can be drawn through any point not on a given line parallel to the given line in a plane."

We've seen that for the equation Y=mX + b the Y intercept will always be at b because that is where X=0.

Using Algebra we can subtract b from both sides: Y - b = mX

and multiply by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {1}{m}}

${\frac {Y}{m}}-{\frac {b}{m}}=X$

we can see that the X intercept is going to be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{b}{m}} .

An axis intercept may simply refer to the number value on the axis where the intersection occurs. For brevity we may say the line has an X intercept of 1 and a Y intercept of 2. After graphing just a few lines you will be able to tell this line points down and runs through quadrants II, I, and IV. With a little more practice you will be able to know that the equation for the line is Y=-2x + 2. We will see that by specifying the two points we are actually implying the slope of the line. There is an exception to this rule. If we say a line crosses the axes at 0 we know that the line will pass through 2 quadrants instead of 3, but we won't know which quadrants or how steep the line is. When we look at slope in the next section we will see why the equations above specify a point and a slope.

When you are trying to graph a linear equation finding the axes intercepts is often the easiest way to go about doing it. To find the x-intercept, set y = 0 and solve for x. To find the y-intercept, set x = 0 and solve for y. For most examples the intercepts are different points, and a line can be drawn through the two intercepts. If both intercepts are (0,0), then another point must be determined to graph the line. If the equations is in the form x = c or y = c, the horizontal or vertical lines are very simple to plot.

Slope

Algebra/Slope

Slope is the change in the vertical distance of a line on a coordinate plane over the change in horizontal difference. In other words, it is the “rise” over the “run” or the steepness of a line. Slope is usually represented by the symbol $m$ like in the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = mx + b } , m the coefficient of x represents the slope of the line.

Slope is computed by measuring the change in vertical distance divided by the change in horizontal difference, i.e.:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ m = \frac{\Delta y}{\Delta x} = \frac{rise}{run} }

The Greek uppercase letter $\Delta$ represents change, in this case change in the y-coordinates divided by the change in x-coordinates.

Positive Slope/ Negative Slope

If a line goes up from left to right, then the slope has to be positive. For example, a slope of ¾ would have a “rise” of 3, or go up 3; and a “run” of 4, or go right 4. Both numbers in the slope are either negative or positive in order to have a positive slope.

If a line goes down from left to right, then the slope has to be negative. For example, a slope of -3/4 would have a “rise” of -3, or go down 3; and a “run” of 4, or go right 4. Only one number in the slope can be negative for a line to have a negative slope.

Other Types of Slope

There are two special circumstances, no slope and slope of zero. A horizontal line has a slope of 0 and a vertical line has an undefined slope.

Horizontal lines have the form: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ y = a }  ; where a is a constant, i.e. $a\in R$
Vertical lines have the form: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ x = a }  ; where a is a constant, i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in R }

Resources

• Solving Linear Equations, Lumen Learning
• General Strategy to Solving Linear Equations, OpenStax
• Solving Linear Equations for a Specific Variable, OpenStax
• Solving Linear Equations: Example 1. Produced by TA Catherine Sporer, UTSA
• Solving Linear Equations: Example 2. Produced by TA Catherine Sporer, UTSA