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In this lesson, it will be shown how a graph can help find values of variables that satisfy two linear equations *simultaneously*.
### Catch-Up and Review

Which graph corresponds to $y=x−1?$

**Here are a few recommended readings before getting started with this lesson.**

- Linear Equation
- Slope-intercept form
- Graphing linear equations in slope-intercept form
- Literal Equations

**Here are a few practice exercises before getting started with this lesson.**

a Write the equation $-6x+2y=12$ in slope-intercept form.

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b Consider the following linear graphs.

{"type":"choice","form":{"alts":["Graph I","Graph II","Graph III","Graph IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}

Challenge

The number of students in the US participating in high school basketball and soccer has steadily increased over the past few years.

The following table shows some information about this.

High School Sport | Number of Students Participating in $2015$ (Thousands) |
Average Rate of Increase (Thousands per Year) |
---|---|---|

Basketball | $700$ | $20$ |

Soccer | $400$ | $40$ |

Consider the above table to answer the following questions.

a Write two equations to represent the situation.

b Graph the equations from Part A in the same coordinate plane. Use the graph to predict the approximate year when the number of students participating in these two sports will be the same.

Explore

In the graph below, four lines and their corresponding linear equations can be seen on a coordinate plane. Investigate which lines intersect at one point, which lines intersect at infinitely many points, and which lines do not intersect at all.

Discussion

Two important concepts will be discussed here.

Concept

An equation in two variables is a mathematical relation between two equal quantities that involves two variables.

Solving an equation in two variables results in an ordered pair that makes the equation true.

Concept

A system of equations is a set of two or more equations involving the same variables. The solutions to a system of equations are values for these variables that satisfy all the equations *simultaneously*. A system of equations is usually written as a vertical list with a curly bracket on the left-hand side.

${2x−3y=13x+y=7 $

Graphically, solutions to systems of equations are the points where the graphs of the equations intersect. For this reason, these solutions are usually expressed as coordinates.
Systems of equations can be solved graphically or algebraically.Discussion

Solving a system of linear equations graphically means graphing the lines represented by the equations of the system and identifying the point of intersection. Consider an example system of equations.
*expand_more*
*expand_more*
*expand_more*
Sometimes the point of intersection of the lines is not a lattice point. In this case, the solution found by solving the system of equations graphically is approximate.

${2y=-2x+8x=y−1 $

To solve the system of equations, three steps must be followed.
1

Write the Equations in Slope-Intercept Form

Start by writing the equations in slope-intercept form by isolating the $y-$variables. For the first linear equation, divide both sides by $2.$ For the second equation, add $1$ to both sides.

${2y=-2x+8x=y−1 (I)(II) $

▼

$(I):$ Solve for $y$

DivEqn

$(I):$ $LHS/2=RHS/2$

${y=2-2x+8 x=y−1 $

WriteSumFrac

$(I):$ Write as a sum of fractions

${y=2-2x +28 x=y−1 $

MovePartNumRight

$(I):$ $ca⋅b =ca ⋅b$

${y=2-2 x+28 x=y−1 $

MoveNegNumToFrac

$(I):$ Put minus sign in front of fraction

${y=-22 x+28 x=y−1 $

CalcQuot

$(I):$ Calculate quotient

${y=-1x+4x=y−1 $

IdPropMult

$(I):$ Identity Property of Multiplication

${y=-x+4x=y−1 $

▼

$(II):$ Solve for $y$

${y=-x+4y=x+1 $

2

Graph the Lines

Now that the equations are both written in slope-intercept form, they can be graphed on the same coordinate plane.

3

Identify the Point of Intersection

The point where the lines intersect is the solution to the system.

The lines appear to intersect at $(1.5,2.5).$ Therefore, this is the solution to the system — the value of $x$ is $1.5$ and the value of $y$ is $2.5.$

Example

Tearrik is throwing a party and bought some sodas and chips.

He bought $21$ items and spent $$51.$ The cost of a soda is $$2$ and the cost of a bag of chips is $$3.$ This can be modeled by a system of equations.${x+y=212x+3y=51 (I)(II) $

Here, $x$ is the number of sodas and $y$ the number of bags of chips Tearrik bought. Solve the system by graphing and find how many sodas and bags of chips he bought. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Sodas <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>:","formTextAfter":null,"answer":{"text":["12"]}}

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Chips <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span>:","formTextAfter":null,"answer":{"text":["9"]}}

Start by writing both linear equations in slope-intercept form.

To start, each equation in the system will be written in slope-intercept form. This means that the variable $y$ will be isolated in both equations.
Now, the slope and the $y-$intercept of each line will be used to draw the graphs on the same coordinate plane. Since the number of items cannot be negative, only the first quadrant will be considered for the graph.

${x+y=212x+3y=51 (I)(II) $

▼

$(I), (II):$ Write in slope-intercept form

SubEqn

$(I):$ $LHS−x=RHS−x$

${y=-x+212x+3y=51 $

SubEqn

$(II):$ $LHS−2x=RHS−2x$

${y=-x+213y=-2x+51 $

DivEqn

$(II):$ $LHS/3=RHS/3$

${y=-x+21y=3-2x+51 $

WriteSumFrac

$(II):$ Write as a sum of fractions

${y=-x+21y=3-2x +351 $

MovePartNumRight

$(II):$ $ca⋅b =ca ⋅b$

${y=-x+21y=3-2 x+351 $

MoveNegNumToFrac

$(II):$ Put minus sign in front of fraction

${y=-x+21y=-32 x+351 $

CalcQuot

$(II):$ Calculate quotient

${y=-x+21y=-32 x+17 $

Finally, the point of intersection $P$ can be identified.

The point of intersection of the lines is $P(12,9).$ In the context of the situation, this means that Tearrik bought $12$ sodas and $9$ bags of chips.

Pop Quiz

Discussion

The system of equations in the previous example had one solution. This leads to two important definitions.

Concept

A system of equations that has *one or more* solutions is called a consistent system. For example, consider the following linear systems.

$Example System I {y=x+1y=2x−2 Example System II {y=x−12y=2x−2 $

To determine the number of solutions, each system can be graphed on a coordinate plane.
In the graph, it can be seen that the first system has exactly one solution. The second system has infinitely many solutions. Therefore, since both systems have one or more solutions, they are consistent systems.

Concept

An independent system is a system of equations with *exactly* one solution. Consider the following linear system.
*exactly* one solution, it is an independent system.

${y=x+1y=2x−2 $

To state the number of solutions, both lines of the system can be graphed on the same coordinate plane.
Since the system has Example

At Tearrik's party, Ignacio found one of Tearrik's homework assignments. He does not want to do Tearrik's homework for him, but Ignacio decides to quiz himself using the assignment.

Help Ignacio determine whether the given systems of equations are consistent systems and whether they are independent systems.

a ${y=-2x+64x−2y=-20 $

{"type":"choice","form":{"alts":["The system is consistent and independent.","The system is neither consistent nor independent.","The system is consistent but not independent.","The system is independent but not consistent."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}

b ${x+y=34x+4y=12 $

{"type":"choice","form":{"alts":["The system is consistent and independent.","The system is neither consistent nor independent.","The system is consistent but not independent.","The system is independent but not consistent."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":2}

a Solve the system by graphing and determine the number of solutions.

b Solve the system by graphing and determine the number of solutions.

a The system will be solved graphically. To do so, both linear equations must to be expressed in slope-intercept form. In the first equation, the variable $y$ is already isolated. Therefore, only the second equation needs to be rewritten.

${y=-2x+64x−2y=-20 (I)(II) $

▼

$(II):$ Write in slope-intercept form

SubEqn

$(II):$ $LHS−4x=RHS−4x$

${y=-2x+6-2y=-4x−20 $

DivEqn

$(II):$ $LHS/(-2)=RHS/(-2)$

${y=-2x+6y=-2-4x−20 $

WriteDiffFrac

$(II):$ Write as a difference of fractions

${y=-2x+6y=-2-4x −-220 $

DivNegNeg

$(II):$ $-b-a =ba $

${y=-2x+6y=24x −-220 $

MovePartNumRight

$(II):$ $ca⋅b =ca ⋅b$

${y=-2x+6y=24 x−-220 $

MoveNegDenomToFrac

$(II):$ Put minus sign in front of fraction

${y=-2x+6y=24 x−(-220 ) $

SubNeg

$(II):$ $a−(-b)=a+b$

${y=-2x+6y=24 x+220 $

CalcQuot

$(II):$ Calculate quotient

${y=-2x+6y=2x+10 $

The lines intersect at exactly one point.

Since the system has *one or more* solutions, it is a consistent system. Furthermore, since it has *exactly one* solution, it is also an independent system.

b Again, the system of equations will be solved graphically. To do so, both equations will be rewritten in slope-intercept form.
Note that both equations are simplified to the same equation. It can be graphed by using its slope and its $y-$intercept.
The lines overlap each other. Therefore, they intersect at infinitely many points. Since the system has *one or more* solutions, it is a consistent system. However, since it does not have *exactly* one solution, it is not an independent system.

${x+y=34x+4y=12 (I)(II) $

▼

$(I), (II):$ Write in slope-intercept form

SubEqn

$(I):$ $LHS−x=RHS−x$

${y=-x+34x+4y=12 $

SubEqn

$(II):$ $LHS−4x=RHS−4x$

${y=-x+34y=-4x+12 $

DivEqn

$(II):$ $LHS/4=RHS/4$

${y=-x+3y=4-4x+12 $

WriteSumFrac

$(II):$ Write as a sum of fractions

${y=-x+3y=4-4x +412 $

MoveNegNumToFrac

$(II):$ Put minus sign in front of fraction

${y=-x+3y=-44x +412 $

MovePartNumRight

$(II):$ $ca⋅b =ca ⋅b$

${y=-x+3y=-44 x+412 $

CalcQuot

$(II):$ Calculate quotient

${y=-x+3y=-1x+3 $

IdPropMult

$(II):$ Identity Property of Multiplication

${y=-x+3y=-x+3 $

Discussion

In the last example, one of the systems had infinitely many solutions. Systems of equations with infinitely many solutions have a special name.

Concept

A dependent system is a system of equations with infinitely many solutions. Consider the following linear system.

${2y=4x−4y=2x−2 $

To find the number of solutions, both lines of the system can be graphed on the same coordinate plane.
The lines overlap each other. Therefore, the system has infinitely many solutions. This means that this is a dependent system.Furthermore, systems with no solutions also have a special name.

Concept

A system of equations that has no solutions is called an inconsistent system. For example, consider the following linear system.

${y=2x−1y=2x+2 $

To determine the number of solutions, the system can be graphed on a coordinate plane.
The lines do not intersect. The system has no solution and is therefore an inconsistent system.Example

Mark cannot go to Tearrik's party because he has recently started working with his father at a car dealership. At the dealership, Mark's father sells sedans and trucks.

Mark modeled the number of sedans and trucks sold with a system of equations.${y=2x+22y−4x=10 (I)(II) $

Here, $x$ represents the number of days since Mark started working in the agency. In Equation (I) the variable $y$ represents the number of sedans sold. Similarly, in Equation (II) $y$ represents the number of trucks sold. Mark wants to interpret this system of equations in terms of consistency and independence. Help him do this! {"type":"multichoice","form":{"alts":["The system is a consistent system.","The system is an inconsistent system.","The system is a dependent system.","The system is an independent system."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":[1]}

Graph both equations on the same coordinate plane.

To interpret the system in terms of consistency and independence, both linear equations will be drawn on the same coordinate plane. To do so, the slope-intercept form will be used. Since the first equation is already written in this form, the $y-$variable will be isolated in the second equation.
Now that both equations are written in slope-intercept form, their slopes and $y-$intercepts can be used to draw the lines.

${y=2x+22y−4x=10 (I)(II) $

▼

$(II):$ Write in slope-intercept form

AddEqn

$(II):$ $LHS+4x=RHS+4x$

${y=2x+22y=4x+10 $

DivEqn

$(II):$ $LHS/2=RHS/2$

${y=2x+2y=24x+10 $

WriteSumFrac

$(II):$ Write as a sum of fractions

${y=2x+2y=24x +210 $

MovePartNumRight

$(II):$ $ca⋅b =ca ⋅b$

${y=2x+2y=24 x+210 $

CalcQuot

$(II):$ Calculate quotient

${y=2x+2y=2x+5 $

Next, recall the definitions of consistent, inconsistent, dependent, and independent systems.

Consistent System | A system of equations that has one or more solutions. |
---|---|

Independent System | A system of equations with exactly one solution. |

Dependent System | A system of equations with infinitely many solutions. |

Inconsistent System | A system of equations that has no solution. |

With this information in mind, the lines will be considered one more time.

Since the lines do not intersect each other, the system has no solution. Therefore, it is an inconsistent system.

Pop Quiz

A system of equations can be consistent or inconsistent. In addition, a consistent system can be independent or dependent. Classify the following systems of equations in terms of consistency and independence.

Closure

With the content learned in this lesson, the challenge presented at the beginning can be solved. It has been said that the number of students in the US participating in high school basketball and soccer has steadily increased over the past few years.

The following table shows some information about this.

High School Sport | Number of Students Participating in $2015$ (Thousands) |
Average Rate of Increase (Thousands per Year) |
---|---|---|

Basketball | $700$ | $20$ |

Soccer | $400$ | $40$ |

Consider the above table to answer the following questions.

a Write two equations in slope-intercept form to represent this situation.

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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Approximate year:","formTextAfter":null,"answer":{"text":["2030"]}}

a For both equations, let $x$ be the number of years since $2015.$

b Pay close attention to the $x-$coordinate of the point of intersection of the lines.

a To start, the equation for the number of students participating in high school basketball will be written. The variables can be defined as follows.

Variable | Meaning of the Variable |
---|---|

$x$ | Number of years since $2015$ |

$y$ | Number of students that participate in high school basketball (thousands) |

$y=20x+700 $

Next, the equation for the number of students participating in high school soccer will be written. The variables can be defined in a similar way as before. Variable | Meaning of the Variable |
---|---|

$x$ | Number of years since $2015$ |

$y$ | Number of students that participate in high school soccer (thousands) |

$y=40x+400 $

b The equations found in Part A form a system of equations.

${y=20x+700y$