Lesson Solving Systems of Equations by Substitution

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Finding intersection points by actual graphing has two disadvantages. It is tedious and it is of limited accuracy. In this lesson, you will learn how to calculate such intersection points.
A pair of equations with the same two variables is called a system of equations. The ordered pair where the graphs intersect each other makes both equations true. It is called the solution of the system.

Example 1
To solve the system
system( 2x+5y=-10, 3x-y=4 )
you can transform one equation so that a variable is by itself. Solving equation 2 for y in terms of x gives you:
-y=-3x+4
Multiply the whole equation by -1 to get rid of the negative y
y=3x-4

Where the graphs intersect, the y in one equation stands for the same number as the y in the other. So you may substitute 3x-4 for the y in Equation 1.

2x+5(3x-4)=-10
The result is an equation with only one variable. Solve it for x.
Distribute the 5
2x+15x-20=-10}}<BR> Combine like terms; add 20 to each number<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=17x=10%22+BORDER=0+%3E%0D%0ADivide+each+side+by+17%0D%0A%7B%7B%7Bx=10%2F17&x=0003 ALIGN=MIDDLE ALT=
Divide each side by 17
x=10/17' BORDER=0 ><BR> If you want decimal form, <BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=x=.5882%22+BORDER=0+%3E%0D%0AI+would+use+fraction+form+because+it+is+easier+to+use.%0D%0A%0D%0A%0D%0ASubstitute+your+answer+for+x+into+the+other+equation+to+get+your+y+value.+%0D%0A%0D%0A%7B%7B%7By=3%2810%2F17%29-4&x=0003 ALIGN=MIDDLE ALT=
I would use fraction form because it is easier to use.

Substitute your answer for x into the other equation to get your y value.
y=3(10/17)-4' BORDER=0 ><BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=y=%2830%2F17%29-4%22+BORDER=0+%3E%0D%0A%7B%7B%7By=-38%2F17&x=0003 ALIGN=MIDDLE ALT=
y=-38/17' BORDER=0 > <BR> <BR> Now you have your solution:<BR> 10/17 and -38/17 <BR> <BR> If you have attempted to solve by graphing, you would end up with this:<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C400%2C-10%2C10%2C-10%2C10%2C3x-4%2C%28-2%2F5%29x-2%29%22+BORDER=0+%3E%0D%0A%0D%0A%0D%0ACHECK+YOUR+ANSWER%3A%0D%0ATo+check%2C+all+you+need+to+do+is+plug+the+values+for+x+and+y+into+one+of+the+equations.++If+the+statement+is+true%2C+your+answers+are+correct.++If+it+is+false%2C+your+answers+are+incorrect.%0D%0A%0D%0A%0D%0A%7B%7B%7B2%2810%2F17%29%2B5%28-38%2F17%29=-10&x=0003 ALIGN=MIDDLE ALT=

CHECK YOUR ANSWER:
To check, all you need to do is plug the values for x and y into one of the equations. If the statement is true, your answers are correct. If it is false, your answers are incorrect.

Finding intersection points by actual graphing has two disadvantages. It is tedious and it is of limited accuracy. In this lesson, you will learn how to calculate such intersection points. <BR> A pair of equations with the same two variables is called a system of equations. The ordered pair where the graphs intersect each other makes both equations true. It is called the solution of the system. <BR> <BR> Example 1<BR> To solve the system<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=system%28+2x%2B5y=-10%2C+3x-y=4+%29+&x=0003 ALIGN=MIDDLE ALT='system( 2x+5y=-10, 3x-y=4 ) ' BORDER=0 > <BR> you can transform one equation so that a variable is by itself. Solving equation 2 for y in terms of x gives you:<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=-y=-3x%2B4&x=0003 ALIGN=MIDDLE ALT='-y=-3x+4' BORDER=0 ><BR> Multiply the whole equation by -1 to get rid of the negative y<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=y=3x-4&x=0003 ALIGN=MIDDLE ALT='y=3x-4' BORDER=0 > <BR> <BR> Where the graphs intersect, the y in one equation stands for the same number as the y in the other. So you may substitute <IMG SRC=/cgi-bin/plot-formula.mpl?expression=3x-4&x=0003 ALIGN=MIDDLE ALT='3x-4' BORDER=0 > for the y in Equation 1. <BR> <BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B5%283x-4%29=-10&x=0003 ALIGN=MIDDLE ALT='2x+5(3x-4)=-10' BORDER=0 > <BR> The result is an equation with only one variable. Solve it for x.<BR> Distribute the 5<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B15x-20=-10%7D%7D%0D%0ACombine+like+terms%3B+add+20+to+each+number%0D%0A%7B%7B%7B17x=10&x=0003 ALIGN=MIDDLE ALT='2x+15x-20=-10}}<BR> Combine like terms; add 20 to each number<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=17x=10%22+BORDER=0+%3E%0D%0ADivide+each+side+by+17%0D%0A%7B%7B%7Bx=10%2F17&x=0003 ALIGN=MIDDLE ALT='17x=10' BORDER=0 ><BR> Divide each side by 17<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=x=10%2F17%22+BORDER=0+%3E%0D%0AIf+you+want+decimal+form%2C+%0D%0A%7B%7B%7Bx=.5882&x=0003 ALIGN=MIDDLE ALT='x=10/17' BORDER=0 ><BR> If you want decimal form, <BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=x=.5882%22+BORDER=0+%3E%0D%0AI+would+use+fraction+form+because+it+is+easier+to+use.%0D%0A%0D%0A%0D%0ASubstitute+your+answer+for+x+into+the+other+equation+to+get+your+y+value.+%0D%0A%0D%0A%7B%7B%7By=3%2810%2F17%29-4&x=0003 ALIGN=MIDDLE ALT='x=.5882' BORDER=0 ><BR> I would use fraction form because it is easier to use. <BR> <BR> Substitute your answer for x into the other equation to get your y value. <BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2810%2F17%29-4%22+BORDER=0+%3E%0D%0A%7B%7B%7By=%2830%2F17%29-4&x=0003 ALIGN=MIDDLE ALT='y=3(10/17)-4' BORDER=0 ><BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=y=%2830%2F17%29-4%22+BORDER=0+%3E%0D%0A%7B%7B%7By=-38%2F17&x=0003 ALIGN=MIDDLE ALT='y=(30/17)-4' BORDER=0 ><BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=y=-38%2F17%22+BORDER=0+%3E%0D%0A%0D%0A%0D%0ANow+you+have+your+solution%3A%0D%0A10%2F17+and+-38%2F17%0D%0A%0D%0A%0D%0AIf+you+have+attempted+to+solve+by+graphing%2C+you+would+end+up+with+this%3A%0D%0A%7B%7B%7Bgraph%28500%2C400%2C-10%2C10%2C-10%2C10%2C3x-4%2C%28-2%2F5%29x-2%29&x=0003 ALIGN=MIDDLE ALT='y=-38/17' BORDER=0 > <BR> <BR> Now you have your solution:<BR> 10/17 and -38/17 <BR> <BR> If you have attempted to solve by graphing, you would end up with this:<BR> <IMG SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C400%2C-10%2C10%2C-10%2C10%2C3x-4%2C%28-2%2F5%29x-2%29%22+BORDER=0+%3E%0D%0A%0D%0A%0D%0ACHECK+YOUR+ANSWER%3A%0D%0ATo+check%2C+all+you+need+to+do+is+plug+the+values+for+x+and+y+into+one+of+the+equations.++If+the+statement+is+true%2C+your+answers+are+correct.++If+it+is+false%2C+your+answers+are+incorrect.%0D%0A%0D%0A%0D%0A%7B%7B%7B2%2810%2F17%29%2B5%28-38%2F17%29=-10&x=0003 ALIGN=MIDDLE ALT='graph(500,400,-10,10,-10,10,3x-4,(-2/5)x-2)' BORDER=0 > <BR> <BR> CHECK YOUR ANSWER:<BR> To check, all you need to do is plug the values for x and y into one of the equations. If the statement is true, your answers are correct. If it is false, your answers are incorrect. <BR> <BR> 2(10/17)+5(-38/17)=-10" BORDER=0 >
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:
Simplify 2*(10/17)+5*(-38/17)=-10
  • Graphical form: 2*(10/17)+5*(-38/17)=-10 simplifies to 0=0
  • Text form: 2*(10/17)+5*(-38/17)=-10 simplifies to 0=0
  • Cartoon (animation) form: simplify_cartoon( 2*(10/17)+5*(-38/17)=-10 )
    simplify_cartoon( 2*(10/17)+5*(-38/17)=-10 )

Detailed explanation:

Look at highlight_red( 2*(highlight_red( 10/17 )) )+5*(-38/17)=-10.
Remove unneeded parentheses around factor highlight_red( 10 ),highlight_red( 1/17 )
It becomes 2*highlight_green( 10 )/highlight_green( 17 )+5*(-38/17)=-10.
Look at 2*10/17+highlight_red( 5*(-highlight_red( 38/17 )) )=-10.
Remove unneeded parentheses around factor highlight_red( 38 ),highlight_red( 1/17 )
It becomes 2*10/17-5*highlight_green( 38 )/highlight_green( 17 )=-10.
Look at highlight_red( 2 )*highlight_red( 10 )/17-5*38/17=-10.
Multiplied numerator integers
It becomes highlight_green( 20 )/17-5*38/17=-10.
Look at 20/17-highlight_red( 5 )*highlight_red( 38 )/17=-10.
Multiplied numerator integers
It becomes 20/17-highlight_green( 190 )/17=-10.
Look at highlight_red( 20/17 )-highlight_red( 190/17 )=-10.
Added fractions or integers together
It becomes -highlight_green( 170/17 )=-10.
Look at -highlight_red( 170 )/highlight_red( 17 )=-10.
Factors 170 and 17 have greatest common factor (GCF) of 17. Reducing fraction.
It becomes -highlight_green( 10 )/highlight_green( 1 )=-10.
Look at -10/highlight_red( 1 )=-10.
Remove extraneous '1' from product highlight_red( 1/1 )
It becomes -10=-10.
Look at -10=highlight_red( -10 ).
Moved these terms to the left highlight_green( 10 )
It becomes -10+highlight_green( 10 )=0.
Look at -highlight_red( 10 )+highlight_red( 10 )=0.
Added fractions or integers together
It becomes highlight_green( 0 )=0.
Result: 0=0

Done!

Your answer is correct!
Hope this helps!
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