A system of equations is a set of two or more equations with the same variables.
The solution to a system of equations is the set of values for the variables that satisfy all the equations simultaneously.
The solution to a system of equations is the intersection of the graphs of the equations.
There are different methods to solve systems of equations:
- Graphical method.
- Substitution method.
- Elimination method.
The graphical method is not always accurate, as it depends on the precision of the graph.
Graphical Method
The graphical method consists of:
- Graphing each equation in the system.
- Finding the intersection point(s) of the graphs.
Solve the following system of equations using the graphical method:
\$\$ \begin{cases} y = 2x + 1 \ y = -x + 4 \end{cases} \$\$
- Graph each equation:
- Find the intersection point: \$(1, 3)\$.
Therefore, the solution to the system is \$x = 1\$ and \$y = 3\$.
Substitution Method
The substitution method consists of:
- Solving one of the equations for one of the variables.
- Substituting the expression obtained in the other equation.
- Solving the resulting equation.
- Substituting the value obtained in the expression from step 1.
Solve the following system of equations using the substitution method:
\$\$ \begin{cases} y = 2x + 1 \ y = -x + 4 \end{cases} \$\$
- Solve the first equation for \$y\$: \$y = 2x + 1\$.
- Substitute the expression obtained in the second equation: \$2x + 1 = -x + 4\$.
- Solve the resulting equation:
\$\$ \begin{aligned} 2x + 1 &= -x + 4 \ 3x &= 3 \ x &= 1 \end{aligned} \$\$
- Substitute the value obtained in the expression from step 1: \$y = 2(1) + 1 = 3\$.
Therefore, the solution to the system is \$x = 1\$ and \$y = 3\$.
Elimination Method
The elimination method consists of:
- Adding or subtracting the equations to eliminate one of the variables.
- Solving the resulting equation.
- Substituting the value obtained in one of the original equations.
Solve the following system of equations using the elimination method:
\$\$ \begin{cases} y = 2x + 1 \ y = -x + 4 \end{cases} \$\$
- Subtract the second equation from the first to eliminate \$y\$: \$(2x + 1) - (-x + 4) = 0\$
- Solve the resulting equation:
\$\$ \begin{aligned} 2x + 1 + x - 4 &= 0 \ 3x - 3 &= 0 \ 3x &= 3 \ x &= 1 \end{aligned} \$\$
- Substitute the value obtained in one of the original equations: \$y = 2(1) + 1 = 3\$.
Therefore, the solution to the system is \$x = 1\$ and \$y = 3\$.
Solve the following system of equations using the substitution method:
\$\$ \begin{cases} 2x + y = 5 \ x - y = 1 \end{cases} \$\$
Solve the following system of equations using the elimination method:
\$\$ \begin{cases} 3x + 2y = 7 \ 2x - y = 4 \end{cases} \$\$
