System of Equations

Pronunciation: /ˈsɪs təm ʌv ɪˈkweɪ ʒənz/ ?
y = x2 - 3
y = 2x + 1
Figure 1: A system of equations

A system of equations is a set of equations that are taken to be simultaneously true. A system of equations is also called simultaneous equations. If a system of equations contains only linear equations is a linear system.

A system of equations may have no solution. This means that there are no values for which all the equations are true at once. A system of equations may have one or more solutions. The solution(s) of the system of equations are any values for which all the equations are simultaneously true.

Visualizing a System of Equations

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Manipulative 1: Simultaneous equations. Created with GeoGebra.

A system of equations can be visualized by graphing the equations. The solution(s) of the system are the points of intersections of the curves. If the graphs of the equations do not intersect, the system has no solutions.

The figure in manipulative 1 shows two simultaneous equations. The parabola is green and the line is blue. The red points are the points of intersection. The values of the red points are the solutions to the system. Click on the line and the parabola and drag them to change the figure.

Discovery

Click on the parabola in manipulative 1 and drag it so that there is only one solution. Click on the parabola and drag it so that there are no solutions. To reset the figure to its initial configuration, click on the reset button (GGB Reset button).

What is the greatest number of solutions that can be found for a system consisting of a line and a parabola?

Solving by Substitution

A system of equations can often be solved by substitution. One variable is replaced by an expression equal to that variable. Take the system of equations
y=x^2-1,y=x+1

StepEquationsDescription
1y=x^2-1,y=x+1These are the simultaneous equations to solve.
2x+1=x^2-1Use the substitution property of equality to substitute x+1 in for y.
3x+1-x-1=x^2-1-x-1Since the resulting equation in step 2 is a quadratic, we will be using the quadratic formula to solve it. The quadratic formula requires a zero be on one side of the equation. Use the additive property of equality to add -x-1 to both sides.
x-x+1-1=x^2-x-1-1Use the associative property of addition to get like terms next to each other.
0+0=x^2-x-2Combine like terms.
0=x^2-x-2Use the additive property of zero to simplify the left side of the equation.
4x=((-b+-square root(b^2-4*a*c))/(2*a) implies x=((-(-1)+-square root((-1)^2-4*1*(-2)))/(2*1)Apply the quadratic formula with a=1, b=-1 and c=-2.
x=((1+-square root((-1)^2-4*1*(-2)))/(2*1)Use the definition of negation to simplify the double negative.
x=((1+-square root(1-4*1*(-2)))/(2*1)Simplify the exponent.
x=((1+-square root(1+8))/2Simplify multiplication and division.
x=((1+-square root(9))/2Simplify addition and subtraction.
x=(1+-3)/2Simplify the square root.
x=(1+3)/2, x=(1-3)/2Split the plus or minus into two equations.
x=4/2, x=(-2)/2Split addition and subtraction.
x=2, x=-1Simplify the fractions. These are the possible values of x.
5y=x+1,x=2Use the substitution property of equality to substitute 2 in for x.
y=2+1Perform the substitution.
y=3Simplify the addition.
(x,y)=(2,3)One solution of the system is the ordered pair (2,3).
6y=x+1,x=-1Use the substitution property of equality to substitute -1 in for x.
y=-1+1Perform the substitution.
y=0Simplify the addition.
(x,y)=(-1,0)One solution of the system is the ordered pair (-1,0).
Table 1: Solving simultaneous equations

More Information

  • McAdams, David. Linear System. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Linear%20System.

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System of Equations. 2009-02-10. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/s/systemofequations.html.

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2009-02-10: Initial version (McAdams, David.)

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