AskSia

Plus

Question #1: Create a system of linear equations with no solutions, one solution...

May 13, 2024

Question #1: Create a system of linear equations with no solutions, one solution, and infinitely many solutions. Explain how you determine the number of solutions without solving them. Then, verify the solutions by solving them, both algebraically and graphically (by free-hand).

Solution by Steps

step 1

To create a system with no solutions, the equations must be parallel and have different y-intercepts

step 2

Example:

x + y = 2

and

x + y = 5

. These lines are parallel and will never intersect

step 3

To verify algebraically, subtract the first equation from the second:

(x + y) - (x + y) = 5 - 2

, which simplifies to

0 = 3

, a contradiction

step 4

To create a system with one solution, the equations must be intersecting lines

step 5

Example:

x + y = 2

and

x - y = 0

. These lines intersect at a single point

step 6

To verify algebraically, solve the system: from

x - y = 0

, we get

x = y

. Substituting into the first equation,

x + x = 2

, so

x = 1

and

y = 1

step 7

To create a system with infinitely many solutions, the equations must be the same line or multiples of each other

step 8

Example:

x + y = 2

and

2x + 2y = 4

. These represent the same line

step 9

To verify algebraically, divide the second equation by 2:

2x + 2y = 4

becomes

x + y = 2

, which is the same as the first equation

step 10

To verify graphically, plot the equations on a coordinate plane. For no solutions, the lines will be parallel. For one solution, the lines will intersect at a point. For infinitely many solutions, the lines will coincide

[question 1] Answer

No solutions:

x + y = 2

x + y = 5

. One solution:

x + y = 2

x - y = 0

. Infinitely many solutions:

x + y = 2

2x + 2y = 4

Key Concept

Determining the number of solutions in a system of linear equations without solving

Explanation

Systems with no solutions have parallel lines with different y-intercepts. Systems with one solution have intersecting lines. Systems with infinitely many solutions have coinciding lines or are multiples of each other.