"The only way to be totally free is through education" - Jose Marti

Study Journal Notes

Solving Systems Graphically:

Definitions you should know:

• System of equations

• Solution (to a system of equations)

• Intersection

• Ordered Pair

• Variable

• Slope-intercept form

Lesson 1:

• In order to solve a system of linear equations graphically, follow these steps:

  • put each equation into slope-intercept form​

  • graph the equations

  • look for the point of intersection

    • the ordered pair that represents the point of intersection is the solution​

• In order to verify a solution, substitute the x and y values from the ordered pair back into BOTH original equations, if the equations are balanced (one side equals the other side) then the solution is verified.

Lesson 2:

• When you are given a problem (a story, or context) and you need to solve it using a system, follow these steps:

  • define the variables (for example, let x = number of hours, let y = total cost)​

  • Separate your problem/story into two related parts and make an equation (in slope-intercept form) for each part

    • the thing that is dependent on x is the slope of your equation, the thing that never changes regardless of your x variable is your y-intercept​

  • Graph your equations, and find the solution (see lesson 1)

  • Create a solution statement to communicate your solution (for example: After two hours, each competitor will have traveled 6 kms)

• NOTE: When you have a problem that talks about money per item and the total number of items, one of your equations will be x+y=total number of items.

Lesson 3:

• There are three types of solutions for linear systems involving two equations:

  • one solution - when both equations have a different slope​ (they intersect once)

  • no solution - when both equations have the same slope but different y-intercepts (the lines will never intersect)

  • infinite solutions - when both equations have the same slope and the same y-intercept (they are the same equation, so they have infinite points of intersection)

• Resources:

  • Illustrations of the three solution types

Lesson 4:

• Review Quiz

• Khan Academy Practice

• Online Graphing Quiz

• Interactive graphing quiz

• Worksheet with answer key

​

Solving Systems Algebraically:

Lesson 1:

• Recall: a system of linear equations is a set of at least two linear equations

  • You should have the same number of equations as you have variables​

• In word problems, always identify the "parts" and the "whole". The "whole" will go by itself on one side of the equal sign, and the "parts" will go on the other side of the equal sign.

  • "parts" add up to make a "whole"​

• In word problems, investigate the question statement to know how to define your variables

  • the question statement often starts with "determine...."​

Lesson 2:

• Know the four steps in substitution:

  • choose the simplest of the equations and isolate a variable​

  • substitute the variable from step 1 into the other equation

  • solve for the remaining variable

  • substitute the solution in step 3 into the equation that was chosen in step 1 and solve for the remaining variable

• ALWAYS verify your answer by substituting your solution into the original equations

• TIP: put a box around your solution (written as an ordered pair), put a check-mark next to your verification once both sides balance

Lesson 3:

• See lesson 2

• TIP: in order to get rid of fractions, multiply your entire equation by the denominator

  • if you can't get rid of fractions, remember that adding and subtracting requires the same denominator, multiplying goes straight across, and dividing requires multiplying by a reciprocal.​

Lesson 4:

• To define the variables, look at the question statement (usually at the end of the word problem). Your variables should relate to what you are trying to determine

• Systems of equations are made be identifying parts (all on one side of the equation) and wholes (all on the opposite side of the equation from the parts)

• Remember that verification should be considered a necessary part of the problem solving process

• ALWAYS finish your word problem with a sentence that answers the initial problem

Lesson 5:

• Know the four steps in elimination:​​

  • make the coefficients for one of your variables the same in both equations (except for maybe the sign) - this may require multiplying all of the terms in one or both of the equations to achieve a lowest common multiple​

  • add or subtract the equations to eliminate one of the variables

    • if the signs are the same, subtract​

    • if the signs are different, add

  • solve the resulting equation to determine the value of one of the variables

  • substitute the solution into either of the original equations to determine the value of the other variable

• ALWAYS verify your answer by substituting your solution into the original equation

• TIP: put a box around your solution (written as an ordered pair), put a check-mark next to your verification once both sides balance

Lesson 6:

• Sometimes, you will have to multiply both equations in order to make the coefficient of one of the variables the same in each equation (except for maybe the sign) - your goal is to identify the lowest common multiple of the given coefficients and transform the equations so that both equations have the LCM as a coefficient to the same variable.

Lesson 7:

• See lesson 4 for some tips on solving word problems

• whenever your word problem involves an object or shape that you can draw, draw it!

Lesson 8:

• During this class, we completed a substitution and elimination scavenger hunt. The notes are a review of all components of lessons 1-7. Remember that you have many options in checking your work (substitution, elimination, graphing).