Exercise 26 Page 376

What does it mean when solving a system of equations results in an identity or a contradiction?

Infinitely many solutions.

Practice makes perfect

To determine how many solutions this system has, we will solve it by substitution. Doing so will result in one of three cases.

Result of solving by substitution	Number of solutions
A value for x and y is determined.	One solution
An identity is found, such as 2=2.	Infinitely many solutions
A contradiction is found, such as 2≠ 3.	No solution

This means that we should solve the given system of equations and make our conclusion based on the result.

When solving a system of equations using substitution, there are three steps.

Isolate a variable in one of the equations.
Substitute the expression for that variable into the other equation and solve.
Substitute this solution into one of the equations and solve for the value of the other variable.

For this exercise, y is already isolated in one equation, so we can skip straight to solving!

y= 12x+3 & (I) 2y-x=6 & (II)

(II): y= 1/2x+3

y= 12x+3 & (I) 2( 12x+3)-x=6 & (II)

(II): Distribute 2

y= 12x+3 & (I) x+6-x=6 & (II)

(II): Subtract term

y= 12x+3 & (I) 6=6 & (II)

Solving this system resulted in an identity. This means that the system has infinitely many solutions.