1.1 Integers and Rational Numbers

www.ck12.org Chapter 12. Systems of Linear EquationsThe slopes of the two equations are the same but the y−intercepts are different; therefore the lines are parallel andthe system has no solutions. This is an inconsistent system.Example 3Determine whether the following system has exactly one solution, no solutions, or an infinite number of solutions.x + y = 33x + 3y = 9SolutionWe must rewrite the equations so they are in slope-intercept formx + y = 3 y = −x + 3 y = −x + 3⇒⇒x + 3y = 9 3y = −3x + 9 y = −x + 3The lines are identical; therefore the system has an infinite number of solutions. It is a dependent system.Determining the Type of System AlgebraicallyA third option for identifying systems as consistent, inconsistent or dependent is to just solve the system and use theresult as a guide.Example 4Solve the following system of equations. Identify the system as consistent, inconsistent or dependent.10x − 3y = 32x + y = 9SolutionLets solve this system using the substitution method.Solve the second equation for y:2x + y = 9 ⇒ y = −2x + 9Substitute that expression for y in the first equation:10x − 3y = 310x − 3(−2x + 9) = 310x + 6x − 27 = 316x = 30x = 158Substitute the value of x back into the second equation and solve for y:263