L5 Solving Systems of Equations - Part 1
Completion requirements
Unit 1A
Precalculus
Lesson 5: Solving Systems of Equations
There are many mathematical situations that can be modelled by a
seriesof equations involving more than one variable. These systems of equations are sets of equations that can be solved simultaneously. Consider an amusement park that has two-tiered admission prices. Children pay «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»$«/mo»«mn»5«/mn»«mo».«/mo»«mn»00«/mn»«/mrow»«/mstyle»«/math» and adults pay «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»$«/mo»«mn»10«/mn»«mo».«/mo»«mn»00«/mn»«/mstyle»«/math».
On a certain summer day, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«/mstyle»«/math» people entered the park and «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»$«/mo»«mn»8«/mn»«mo»§#160;«/mo»«mn»750«/mn»«/mrow»«/mstyle»«/math» was collected from admissions. The number of children and adults that entered the park that day can be
determined by setting up the equations below.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«/mrow»«/mstyle»«/math»number of children.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«/mrow»«/mstyle»«/math»number of adults.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«/mrow»«/mstyle»«/math»number of children.
Let «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«/mrow»«/mstyle»«/math»number of adults.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»§#160;«/mo»«mi».....«/mi»«mo»§#160;«/mo»«mi»Equation«/mi»«mo
mathvariant=¨italic¨»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»A«/mi»«mo»§#160;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»+«/mo»«mn»10«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mo»§#160;«/mo»«mn»750«/mn»«mo»§#160;«/mo»«mi».....«/mi»«mo»§#160;«/mo»«mi»Equation«/mi»«mo
mathvariant=¨italic¨»§#160;«/mo»«mi mathvariant=¨normal¨»B«/mi»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
A linear system of equations, like the system of two equations above, only involves linear equations. There are three different methods that can be used to solve these systems.
The Example that follows shows how the linear system of equations defined above can be solved using each of the three methods.
- Graphing
- Substitution
- Elimination
The Example that follows shows how the linear system of equations defined above can be solved using each of the three methods.
Solve the system of linear equations defined in the Introduction of this Lesson by
Graph each of the functions and find the point of intersection.
The solution to a system of linear equations occurs at the point of intersection of the graphs of the related functions. By examining the graph, the point of intersection is identified as the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»250«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»750«/mn»«/mrow»«/mfenced»«/mstyle»«/math». Given the variables had been defined as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«/mrow»«/mstyle»«/math» number of children and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«/mstyle»«/math» number of adults, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»250«/mn»«/mstyle»«/math» children and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»750«/mn»«/mstyle»«/math» adults entered the amusement park that day.
a.
graphing,
b.
substitution, and
c.
elimination.
a.
Graphing
In this method, the equations are rewritten as functions of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/mrow»«/mstyle»«/math» and then graphed to find the point of intersection of the two lines. The point of intersection represents the solution to the system of equations.
In this method, the equations are rewritten as functions of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/mrow»«/mstyle»«/math» and then graphed to find the point of intersection of the two lines. The point of intersection represents the solution to the system of equations.
| Recall the slope-intercept form of a linear function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»m«/mi»«/mstyle»«/math» is the slope and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»b«/mi»«/mstyle»«/math» is the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept. |
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mo»-«/mo»«mi»x«/mi»«mo»+«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»§#160;«/mo»«mi».....«/mi»«mo»§#160;«/mo»«mi»Equation«/mi»«mo
mathvariant=¨italic¨»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»A«/mi»«mo»§#160;«/mo»«/mstyle»«/math»
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi»x«/mi»«mo»+«/mo»«mn»875«/mn»«mo»§#160;«/mo»«mi».....«/mi»«mo»§#160;«/mo»«mi»Equation«/mi»«mo
mathvariant=¨italic¨»§#160;«/mo»«mi mathvariant=¨normal¨»B«/mi»«/mstyle»«/math»
Graph each of the functions and find the point of intersection.
The solution to a system of linear equations occurs at the point of intersection of the graphs of the related functions. By examining the graph, the point of intersection is identified as the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»250«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»750«/mn»«/mrow»«/mfenced»«/mstyle»«/math». Given the variables had been defined as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«/mrow»«/mstyle»«/math» number of children and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»=«/mo»«/mstyle»«/math» number of adults, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»250«/mn»«/mstyle»«/math» children and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»750«/mn»«/mstyle»«/math» adults entered the amusement park that day.
b.
Substitution
The substitution method requires the substitution of one equation into the other equation.
The substitution method requires the substitution of one equation into the other equation.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»§#160;«/mo»«mi».....«/mi»«mo»§#160;«/mo»«mi»Equation«/mi»«mo
mathvariant=¨italic¨»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»A«/mi»«mo»§#160;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»+«/mo»«mn»10«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mo»§#160;«/mo»«mn»750«/mn»«mo»§#160;«/mo»«mi».....«/mi»«mo»§#160;«/mo»«mi»Equation«/mi»«mo
mathvariant=¨italic¨»§#160;«/mo»«mi mathvariant=¨normal¨»B«/mi»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Step 1:
Rewrite one equation so it can be substituted into the other equation. Whenever possible, isolate a variable in the equation in which that variable has as a coefficient of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math». In this example, rewrite Equation A and isolate either «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»
or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».
Rearranging Equation A for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» results in the following equation.
Rearranging Equation A for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» results in the following equation.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»-«/mo»«mi»y«/mi»«/mrow»«/mstyle»«/math»
Step 2:
Substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»-«/mo»«mi»y«/mi»«/mstyle»«/math» in place of «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» into the second equation, Equation B. Solve for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mrow»«mn»5«/mn»«mo»(«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»-«/mo»«mi»y«/mi»«mo»)«/mo»«mo»+«/mo»«mn»10«/mn»«mi»y«/mi»«/mrow»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mo»§#160;«/mo»«mn»750«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»5«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»-«/mo»«mn»5«/mn»«mi»y«/mi»«mo»+«/mo»«mn»10«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mo»§#160;«/mo»«mn»750«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»5«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«mo»§#160;«/mo»«mn»750«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»750«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
| Replace «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» in Equation B with «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»-«/mo»«mi»y«/mi»«/mrow»«/mstyle»«/math» from Equation A. Then, solve for the remaining variable «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math». |
Step 3:
Find the value of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math». Substitute «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»750«/mn»«/mrow»«/mstyle»«/math» into the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»-«/mo»«mi»y«/mi»«/mrow»«/mstyle»«/math».
The solution to this system of equations is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»250«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»750«/mn»«/mrow»«/mfenced»«/mstyle»«/math». The solution is identical to the result obtained using the graphical method.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»§#8722;«/mo»«mi»y«/mi»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»§#8722;«/mo»«mn»750«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«mo»=«/mo»«mn»250«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The solution to this system of equations is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»250«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»750«/mn»«/mrow»«/mfenced»«/mstyle»«/math». The solution is identical to the result obtained using the graphical method.
c.
Elimination
The elimination method involves adding or subtracting equations to eliminate one variable, and then solving for the other variable.
The elimination method involves adding or subtracting equations to eliminate one variable, and then solving for the other variable.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«mspace
width=¨0.33em¨/»«mn»000«/mn»«mo»§#160;«/mo»«mo».....«/mo»«mo»§#160;«/mo»«mi»Equation«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»A«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«/mtd»«/mtr»«mtr»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»+«/mo»«mn»10«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mspace
width=¨0.33em¨/»«mn»750«/mn»«mo»§#160;«/mo»«mo».....«/mo»«mo»§#160;«/mo»«mi»Equation«/mi»«mo»§#160;«/mo»«mi mathvariant=¨normal¨» «/mi»«mi mathvariant=¨normal¨»B«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Step 1:
Multiply one or both equations by a constant so the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» or the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-variables in both equations have the same coefficient. For example, if every term in Equation A is multiplied by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math», the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-variables can be eliminated when the equations
are subtracted. It is also possible to multiple Equation A by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» to eliminate «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math», or divide Equation B by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math»
to eliminate «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»5«/mn»«mo»§#160;«/mo»«mn»000«/mn»«mo»§#160;«/mo»«mi».....«/mi»«mo»§#160;«/mo»«mi»Equation«/mi»«mspace
width=¨0.125em¨/»«mi mathvariant=¨normal¨»A«/mi»«/mtd»«/mtr»«mtr»«mtd»«mn»5«/mn»«mi»x«/mi»«mo»+«/mo»«mn»10«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mo»§#160;«/mo»«mn»750«/mn»«mo»§#160;«/mo»«mi».....«/mi»«mo»§#160;«/mo»«mi»Equation«/mi»«mspace
width=¨0.125em¨/»«mspace width=¨0.125em¨/»«mi mathvariant=¨normal¨»B«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Step 2:
Subtract Equation B from Equation A to eliminate «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math», and then solve for «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨right¨ mathsize=¨14px¨»«menclose notation=¨bottom¨»«mtable columnspacing=¨0px default default default default¨ columnalign=¨center
right center right left left¨»«mtr»«mtd/»«mtd»«mn»5«/mn»«mi»x«/mi»«/mtd»«mtd»«mo»+«/mo»«/mtd»«mtd»«mn»5«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#160;«/mo»«mn»5000«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»-«/mo»«/mtd»«mtd»«mo»§#160;«/mo»«mo»§#160;«/mo»«mn»5«/mn»«mi»x«/mi»«/mtd»«mtd»«mo»+«/mo»«/mtd»«mtd»«mn»10«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»§#160;«/mo»«mn»8750«/mn»«/mtd»«/mtr»«/mtable»«/menclose»«mspace
linebreak=¨newline¨/»«mtable columnspacing=¨default default default default 0px¨ columnalign=¨center center center right center right¨»«mtr»«mtd/»«mtd/»«mtd/»«mtd»«mo»-«/mo»«mn»5«/mn»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mn»3750«/mn»«/mtd»«/mtr»«mtr»«mtd/»«mtd/»«mtd/»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»750«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Step 3:
Substitute the value for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» into original Equation A, and solve for «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».
The solution to this system of equations is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»250«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»750«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
The solution is identical to the result obtained using the graphical and substitution methods.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»750«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»250«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The solution to this system of equations is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»250«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»750«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
The solution is identical to the result obtained using the graphical and substitution methods.
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