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Unit 4 – Linear Systems in Two Dimensions

Unit 4 – Linear Systems in Two Dimensions. Topic: Solving Linear Systems of Equations. Identifying Solutions of Systems. Determine if the ordered pair is an element of the solution set for the given system. (4, 1); {.

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Unit 4 – Linear Systems in Two Dimensions

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  1. Unit 4 – Linear Systems in Two Dimensions Topic: Solving Linear Systems of Equations

  2. Identifying Solutions of Systems • Determine if the ordered pair is an element of the solution set for the given system. (4, 1); { Substitute the values into each equation to see if you get true statements. Both equations are true, thus (4, 1) is an element of the solution set.

  3. Types of systems • Independent system • Solution set is a single point on the coordinate plane. • Graph contains intersecting lines (equations have different slopes); solution is ALWAYS a coordinate point. • Dependent system • System with an infinite solution set. • Graph contains coinciding lines (one line on top of the other; both equations have same slope & y-intercept). • Inconsistent system • System with an empty solution set (no solutions). • Graph contains parallel lines (same slope, different y-intercepts).

  4. Classifying Systems • Classify the system & determine the number of solutions. { Rewrite the first equation in slope-intercept form. Equations have the same slope but different y-intercepts, so lines are parallel. Inconsistent system. No solutions.

  5. Classifying Systems • Classify the system & determine the number of solutions. { Rewrite each equation in slope-intercept form. Equations have different slopes, so lines will intersect. Independent system. One solution.

  6. Solving Systems of Equations by Graphing • Graph each equation • The point where the two lines intersect is the solution to the system. • ALWAYS check your solution algebraically (plug intersection point into both equations to see if you get a true statement). • Easiest to use when both equations are already in slope-intercept form.

  7. Solving Systems of Equations by Graphing { Graph each line and find the intersection. The lines appear to intersect at (4, 0). Check solution algebraically by substituting (4, 0) into each equation.

  8. Solving Systems of Equations by Graphing { Both equations are true, thus (4, 0) is the solution to the system.

  9. Solving Systems of Equations by Substitution • Solve one equation for one variable & substitute that expression into the other equation. • Easiest to use when one equation is already solved for a variable.

  10. { This equation is “solved” for y. Substitute this expression for y into the other equation. Solve this equation for x. Substitute this value into one of the original equations and solve for y. Solving Systems of Equations by Substitution

  11. { Solving Systems of Equations by Substitution The solution to the system is (3, -1).

  12. Solving Systems of Equations by Elimination • Combining the two equations in a way that eliminates one of the variables. • Easiest to use when one variable in each equation has the same coefficient.

  13. { Solve the resulting equation for y. Substitute this value into one of the original equations and solve for x. Solving Systems of Equations by Elimination Both equations have an x coefficient of 1. Subtract one equation from the other to eliminate x.

  14. { Solving Systems of Equations by Elimination The solution to the system is (2, 1).

  15. JOURNAL ENTRY • TITLE: Checking My Understanding: Solving Linear Systems of Equations • Review your notes from this presentation & create and complete the following subheadings: • “Things I already knew:” Identify any information with which you were already familiar. • “New things I learned:” Identify any new information that you now understand. • “Questions I still have:” What do you still want to know or do not fully understand?

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