Lecture Notes
Applied Mathematics for
Business, Economics, and the
Social Sciences (4th Edition);
by Frank S. Budnick
, Chapter 2: Linear Equations
Definition: Linear equations are first degree equations. Each
variable in the equation is raised to the first power.
Definition:
A linear equation involving two variables x and y has the
standard form
ax + by= c
where a, b and c are constants and a and b cannot both equal
zero.
Note: The presence of terms having power other than 1 or
product of variables, e.g. (xy) would exclude an equation from
being linear. Name of the variables may be different from x and
y.
Examples:
1. 3𝑥 + 4𝑦 = 7 is linear equation, where 𝑎 = 3, 𝑏 = 4, 𝑐 = 7
2. √𝑥 = 5 + 𝑦 is non-liner equation as power of 𝑥 is not 1.
Solution set of an equation
Given a linear equation ax + by= c, the solution set for the
equation (2.1) is the set of all ordered pairs (x, y) which satisfy
the equation.
𝑆 = {(𝑥, 𝑦)|𝑎𝑥 + 𝑏𝑦 = 𝑐}
For any linear equation, S consists of an infinite number of
elements.
Method
1. Assume a value of one variable
, 2. Substitute this into the equation
3. Solve for the other variable
Example
Given 2𝑥 + 4𝑦 = 16 , determine the pair of values which
satisfy the equation when x=-2
Solution: Put x=-2 in given equation gives us 4y=16-4, i.e y=3.
So the pair (-2,3) is a pair of values satisfying the given
equation.
Linear equation with n variables
Definition
A linear equation involving n variables x1, x2, . . . , xn has the
general form
a1 x 1 + a2 x 2 + . . . + an x n = b
where 𝑎1, 𝑎2, ⋯ , 𝑎𝑛 are non-zero.
Definition:
The solution set S of a linear equation with n variables as
defined above is the collection of n-tuples (𝑥1, 𝑥2, ⋯ , 𝑥𝑛) such
that 𝑆 = {(𝑥1, ⋯ , 𝑥𝑛)| 𝑎1𝑥1 + ⋯ + 𝑎𝑛𝑥𝑛 = 𝑏}.
As in the case of two variables, there are infinitely many values
in the solution set.
Example
Given an equation 4𝑥1 − 2𝑥2 + 6𝑥3 = 0, what values satisfy
the equation when 𝑥1 = 2 and 𝑥3 = 1.
, Solution: Put the given values of 𝑥1 and 𝑥3 in the above
equation gives 𝑥2 = 7. Thus (2,7,1) is a solution of the above
equation.
Graphing two variable equations
A linear equation involving two variables graphs as a straight
line in two dimensions.
Method:
1. Set one variable equal to zero
2. Solve for the value of other variable
3. Set second variable equal to zero
4. Solve for the value of first variable
5. The ordered pairs (0, y) and (x, 0) lie on the line
6. Connect these points and extend the line in both
directions.
Example
Graph the linear equation 2𝑥 + 4𝑦 = 16
Solution: In lectures
x-intercept
The x-intercept of an equation is the point where the graph of
the equation crosses the x-axis, i.e. y=0.
y- intercept
The y-intercept of an equation is the point where the graph of
the equation crosses the y-axis, i.e. x=0
Note: Equations of the form x=k has no y-intercept and
equations of the form y=k has no x-intercept
Applied Mathematics for
Business, Economics, and the
Social Sciences (4th Edition);
by Frank S. Budnick
, Chapter 2: Linear Equations
Definition: Linear equations are first degree equations. Each
variable in the equation is raised to the first power.
Definition:
A linear equation involving two variables x and y has the
standard form
ax + by= c
where a, b and c are constants and a and b cannot both equal
zero.
Note: The presence of terms having power other than 1 or
product of variables, e.g. (xy) would exclude an equation from
being linear. Name of the variables may be different from x and
y.
Examples:
1. 3𝑥 + 4𝑦 = 7 is linear equation, where 𝑎 = 3, 𝑏 = 4, 𝑐 = 7
2. √𝑥 = 5 + 𝑦 is non-liner equation as power of 𝑥 is not 1.
Solution set of an equation
Given a linear equation ax + by= c, the solution set for the
equation (2.1) is the set of all ordered pairs (x, y) which satisfy
the equation.
𝑆 = {(𝑥, 𝑦)|𝑎𝑥 + 𝑏𝑦 = 𝑐}
For any linear equation, S consists of an infinite number of
elements.
Method
1. Assume a value of one variable
, 2. Substitute this into the equation
3. Solve for the other variable
Example
Given 2𝑥 + 4𝑦 = 16 , determine the pair of values which
satisfy the equation when x=-2
Solution: Put x=-2 in given equation gives us 4y=16-4, i.e y=3.
So the pair (-2,3) is a pair of values satisfying the given
equation.
Linear equation with n variables
Definition
A linear equation involving n variables x1, x2, . . . , xn has the
general form
a1 x 1 + a2 x 2 + . . . + an x n = b
where 𝑎1, 𝑎2, ⋯ , 𝑎𝑛 are non-zero.
Definition:
The solution set S of a linear equation with n variables as
defined above is the collection of n-tuples (𝑥1, 𝑥2, ⋯ , 𝑥𝑛) such
that 𝑆 = {(𝑥1, ⋯ , 𝑥𝑛)| 𝑎1𝑥1 + ⋯ + 𝑎𝑛𝑥𝑛 = 𝑏}.
As in the case of two variables, there are infinitely many values
in the solution set.
Example
Given an equation 4𝑥1 − 2𝑥2 + 6𝑥3 = 0, what values satisfy
the equation when 𝑥1 = 2 and 𝑥3 = 1.
, Solution: Put the given values of 𝑥1 and 𝑥3 in the above
equation gives 𝑥2 = 7. Thus (2,7,1) is a solution of the above
equation.
Graphing two variable equations
A linear equation involving two variables graphs as a straight
line in two dimensions.
Method:
1. Set one variable equal to zero
2. Solve for the value of other variable
3. Set second variable equal to zero
4. Solve for the value of first variable
5. The ordered pairs (0, y) and (x, 0) lie on the line
6. Connect these points and extend the line in both
directions.
Example
Graph the linear equation 2𝑥 + 4𝑦 = 16
Solution: In lectures
x-intercept
The x-intercept of an equation is the point where the graph of
the equation crosses the x-axis, i.e. y=0.
y- intercept
The y-intercept of an equation is the point where the graph of
the equation crosses the y-axis, i.e. x=0
Note: Equations of the form x=k has no y-intercept and
equations of the form y=k has no x-intercept
