Unit 3: systems of equations and inequalities Final Exam Review

Topics to Include Systems of Equations (2 lines) Line and Quadratic Line and Circle Graphing Systems of Equations Graphing Inequalities Systems of Inequalities

Systems of Equations To solve a system of equations, you can use one of 2 strategies SUBSTITUTION With Substitution, look for ONE VARIABLE that is by itself. Plug that variable back into the SECOND EQUATION and solve for the other variable. ELIMINATION With Elimination, your goal is to get one variable to CANCEL OUT To do this, transform the system so that the variable you want to cancel is POSITIVE in one equation and NEGATIVE in the other.

Systems of Equations Example of Substitution Example of Elimination

Systems of Equations You Try SUBSTITUTION 𝒚=𝟔𝒙−𝟏𝟏 −𝟐𝒙−𝟑𝒚=−𝟕 You Try ELIMINATION 𝟓𝒙+𝒚=𝟗 𝟏𝟎𝒙 –𝟕𝒚=−𝟏𝟖

Line and Quadratic Use SUBSTITUTION to solve systems with lines and quadratics Set both equations EQUAL to each other FACTOR and Solve. Make sure to plug the x value back in to get the y value. Example 𝒚= 𝒙 𝟐 +𝟐𝒙+ 𝟒 𝒙 𝟐 +𝟐𝒙+ 𝟒=𝟔𝒙+𝟏 𝒚=𝟔𝒙+𝟏 𝒙 𝟐 −𝟒𝒙+𝟑=𝟎 𝒙−𝟑 𝒙−𝟏 =𝟎 𝒙=𝟑, 𝒙=𝟏 Answers: (3, 19) and (1, 8)

Line and Quadratic You Try 𝒚= 𝒙 𝟐 −𝟒𝒙−𝟓 𝒚= −𝒙−𝟕

Line and Circle Use SUBSTITUTION to solve systems with lines and quadratics FACTOR and Solve. Make sure to plug the x value back in to get the y value. Example 𝒙 𝟐 + 𝒚 𝟐 =𝟗𝟖 𝒙 𝟐 + (𝒙) 𝟐 =𝟗𝟖 𝒚=𝒙 𝟐 𝒙 𝟐 =𝟗 𝒙 𝟐 =𝟒𝟗 𝒙=𝟕 𝒂𝒏𝒅−𝟕 Answers: (7, 7) and (-7, -7)

Line and Circle You Try 𝒙 𝟐 + 𝒚 𝟐 =𝟒𝟓 𝒚=𝟐𝒙

Graphing Systems of Equations Besides Substitution and Elimination, you can also GRAPH the 2 equations to solve Once you graph, the answer is the INTERSECTION of the 2 lines. Example

Graphing Systems of Equations y= -2x+4 Graphing Systems of Equations You Try 1. 𝒚=𝟒𝒙+𝟑 2. 𝒚= 𝒙 𝟐 −𝟒𝒙+𝟒 3. 𝒙 𝟐 + 𝒚 𝟐 =𝟗 𝒚= −𝒙−𝟐 𝒚= −𝟐𝒙+𝟒 𝒚= 𝒙+𝟑

Graphing Inequalities When graphing inequalities on a coordinate plane you must pay attention to 2 things TYPE OF LINE <𝑜𝑟 > : DOTTED LINE ≤𝑜𝑟 ≥ : SOLID LINE SHADING <𝑜𝑟 ≤ : shade BELOW >𝑜𝑟 ≥ : shade ABOVE

Graphing Inequalities Example:

Graphing Inequalities You try: 𝒚< 𝒙 𝟐 −𝟒𝒙+𝟒 𝒚 ≥ 𝟒𝒙 𝟐 −𝟏

Graphing systems of Inequalities When graphing systems of inequalities, graph BOTH lines on the same graph and SHADE both. The answer is where BOTH graphs are shaded. Example

Graphing systems of Inequalities You try: 1. 𝒚 ≤ 𝒙+𝟒 2. 𝒚≥𝒙 𝟐 −𝟓 𝒚< −𝟐𝒙+𝟐 𝒚< −𝒙 𝟐 +𝟏

ALL DONE