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Some real-life situations can be modeled with more than one equation that can be joined into a *system of equations*. There are different methods of solving systems of equations. This lesson will focus on one of them, which involves graphing the equations.
### Catch-Up and Review

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

Explore

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

Discussion

Consider the definition of equations in two variables.

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.

Discussion

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*
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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

Mark is throwing a party, so he bought some donuts and lollipops for his friends.

The number of donuts $x$ and lollipops $y$ that Mark has bought can be modeled by a system of equations.${x+y=203x+2y=52 $

Solve the system by graphing and find how many donuts and lollipops 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":"Donuts <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":"Lollipops <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":["8"]}}

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

To start, each equation in the system will be rewritten in slope-intercept form. This means that the $y-$variable 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. The $y-$intercept of the first equation is $20,$ so the first point is $(0,20).$ The slope is $-1,$ so the second point can be plotted by $1$ step to the right and $1$ step down on the graph. Since this graph uses a different scale, try moving $4$ steps right and $4$ steps down instead.

${x+y=203x+2y=52 (I)(II) $

▼

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

SubEqn

$(I):$ $LHS−x=RHS−x$

${y=-x+203x+2y=52 $

SubEqn

$(II):$ $LHS−3x=RHS−3x$

${y=-x+202y=-3x+52 $

DivEqn

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

${y=-x+20y=2-3x+52 $

WriteSumFrac

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

${y=-x+20y=2-3x +252 $

MovePartNumRight

$(II):$ $ca⋅b =ca ⋅b$

${y=-x+20y=2-3 x+252 $

MoveNegNumToFrac

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

${y=-x+20y=-23 x+252 $

CalcQuot

$(II):$ Calculate quotient

${y=-x+20y=-1.5x+26 $

Since the number of items cannot be negative, only the first quadrant will be considered for the graph. The $y-$intercept of the second equation is $26,$ so the first point on that line is $(0,26).$ The slope is $-1.5.$ To better match the scale of this graph, the second point can be plotted by going $4$ steps to the right and $4⋅1.5=6$ steps down.

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

The point of intersection of the lines is $P(12,8).$ In the context of the situation, this means that Mark bought $x=12$ donuts and $y=8$ lollipops.

Pop Quiz

Consider the graph of a system of equations consisting of two lines. What is the solution to the system?

Discussion

When a system of linear equations has two equations and two variables, the system can have zero, one, or infinitely many solutions.

If a system has no solution, its graph might look similar to this one.

Recall that the solution to a system is the point where the lines intersect. If a system has no solution, then the lines never intersect. In fact, the lines must be parallel, meaning that they have the same slope and different $y$-intercepts. Here is an example of one such system.${y=3x+2y=3x−5 $

If a system of equations has one solution, its graph consists of two lines that intersect exactly once. The point of intersection is the solution to the system.

In contrast to parallel lines, lines that intersect once must have different slopes. For example, the following system must have exactly one solution because the two lines have different slopes.${y=-x+5y=3x−2 $

If a system of equations has infinitely many solutions, the lines intersect at infinitely many points. This means the lines lie on top of each other or coincide with each other.

These lines are said to be coincidental, and since they have the same slope and $y-$intercept, they are different versions of the same line. Here is one example of a system that has an infinite number of solutions.

${y=3x+12y=6x+2 $

Example

To start the party, Mark suggested they play a fun game. He invited everyone to a room full of balloons. Each balloon had a note with a task to complete or a question to answer.
### Hint

### Solution

The number of tasks $t$ and questions $q$ can be determined by the following system of equations.

${5t+8q=11612.5t=290−20q $

Solve the system of equations by graphing. {"type":"text","form":{"type":"equation","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"","description":[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.61508em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">t<\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">q<\/span><\/span><\/span><\/span>"}]},"formTextBefore":null,"formTextAfter":null,"answer":{"text":[{"id":0,"text":""},{"id":1,"text":""}],"noSolution":false,"infiniteSolution":true}}

First, rewrite the equations into slope-intercept form. Then, graph the equations and find the point of intersection of the lines.

To solve the system of equations graphically, both equations should be rewritten in slope-intercept form first. Let $q$ be the independent variable and $t$ be the dependent variable.
Notice that both equations simplify to the same equation. This equation can be graphed by using its slope and its $y-$intercept.

${5t+8q=11612.5t=290−20q (I)(II) $

▼

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

SubEqn

$(I):$ $LHS−8q=RHS−8q$

${5t=-8q+11612.5t=290−20q $

DivEqn

$(I):$ $LHS/5=RHS/5$

${t=5-8q +5116 12.5t=290−20q $

CommutativePropAdd

$(II):$ Commutative Property of Addition

${t=5-8q +5116 12.5t=-20q+290 $

DivEqn

$(II):$ $LHS/12.5=RHS/12.5$

${t=5-8q +5116 t=12.5-20q +12.5290 $

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

${t=-58q +5116 t=-12.520q +12.5290 $

$(I), (II):$ $ca⋅b =ca ⋅b$

${t=-58 q+5116 t=-12.520 q+12.5290 $

$(I), (II):$ Calculate quotient

${t=-1.6q+23.2t=-1.6q+23.2 $

The lines overlap each other. They intersect at infinitely many points, which means that the system of equations has infinitely many solutions. This indicates that the system of equations was not set up properly. Maybe the same information was written in two different ways, which resulted in two equations that represent the same line.

Example

Later in the evening, Mark suggested doing a scavenger hunt. All the hunters were split into two teams. The team that found the most items would win.
Team $1,$ the Adventure Seekers, found $a$ items. Team $2,$ the Mystery Solvers, found $m$ items. Mark’s parents tallied up the scores and, to make it more of a challenge, decided to give the scores as a system of equations. Whoever solves the system will get an extra gift! ### Hint

### Solution

External credits: @upklyak

${9a+4m=542m=34−4.5a $

Solve the system using the graphing method to find how many items each team found. {"type":"text","form":{"type":"equation","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"","description":[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">a<\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">m<\/span><\/span><\/span><\/span>"}]},"formTextBefore":null,"formTextAfter":null,"answer":{"text":[{"id":0,"text":""},{"id":1,"text":""}],"noSolution":true,"infiniteSolution":false}}

Begin by rewriting the equations into slope-intercept form. Then, graph both equations on the same coordinate plane using their slopes and $y-$intercepts.

The given system of equations needs to be solved graphically.
Notice that the equations have the same slope of $-2.25$ but different $y-$intercepts. This means that the lines are parallel. Graph them using the slope and the $y-$intercepts.
**no solution** to the system of equations. Since Mark's parents are not mathematicians, they probably made a mistake, so their system did not work.

${9a+4m=542m=34−4.5a $

To do so, both equations should be rewritten in slope-intercept form first. Let $a$ be the independent variable and $m$ be the dependent variable.
${9a+4m=542m=34−4.5a (I)(II) $

▼

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

SubEqn

$(I):$ $LHS−9a=RHS−9a$

${4m=-9a+542m=34−4.5a $

CommutativePropAdd

$(II):$ Commutative Property of Addition

${4m=-9a+542m=-4.5a+34 $

DivEqn

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

${m=4-9a+54 2m=-4.5a+34 $

DivEqn

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

${m=4-9a+54 m=2-4.5a+34 $

$(I), (II):$ Write as a sum

${m=4-9a +454 m=2-4.5a +234 $

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

${m=-49a +454 m=-24.5a +234 $

$(I), (II):$ Calculate quotient

${m=-2.25a+13.5m=-2.25a+17 $

Recall the the solution to the system of equations is the point of intersection of the lines. However, since the lines are parallel, they never intersect. Therefore, there is

Pop Quiz

Consider the given system of equations. Does it have zero, one, or infinitely many solutions?

Closure

This lesson focused on solving systems of linear equations by graphing. When equations in the system are written in standard or point-slope form, it is unclear how many solutions the system has. However, the situation is different when it comes to slope-intercept form.

These facts can be useful when solving systems of equations.

$Slope-Intercept Formy=mx+b $

If both equations of the system of equations are in slope-intercept form, there is no need to graph them in order to find the number of solutions. It can be found by analyzing the slopes and $y-$intercepts of the equations. Characteristics | Number of Solutions |
---|---|

Same slope and same $y-$intercept | Infinitely many solutions |

Same slope and different $y-$intercepts | No solution |

Different slopes | One solution |

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