
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. 

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. DiscoveryClick 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 (). What is the greatest number of solutions that can be found for a system consisting of a line and a parabola? 
Step  Equations  Description 

1  These are the simultaneous equations to solve.  
2  Use the substitution property of equality to substitute x+1 in for y.  
3  Since 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 x1 to both sides.  
Use the associative property of addition to get like terms next to each other.  
Combine like terms.  
Use the additive property of zero to simplify the left side of the equation.  
4  Apply the quadratic formula with a=1, b=1 and c=2.  
Use the definition of negation to simplify the double negative.  
Simplify the exponent.  
Simplify multiplication and division.  
Simplify addition and subtraction.  
Simplify the square root.  
Split the plus or minus into two equations.  
Split addition and subtraction.  
Simplify the fractions. These are the possible values of x.  
5  Use the substitution property of equality to substitute 2 in for x.  
Perform the substitution.  
Simplify the addition.  
One solution of the system is the ordered pair (2,3).  
6  Use the substitution property of equality to substitute 1 in for x.  
Perform the substitution.  
Simplify the addition.  
One solution of the system is the ordered pair (1,0).  
Table 1: Solving simultaneous equations 
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  Y 
Z  X 
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