Equations with Two Variables
This lesson discusses solving a system of linear equations by substitution, elimination, and graphing, as well as solving a simple system of a linear and a quadratic equation.
Solving a System of Equations by Substitution
A system of linear equations is a set of two or more linear equations in the same variables. A solution to the system is an ordered pair that is a solution in all the equations in the system. The ordered pair (1, -2) is a solution for the system of equations Â\(2x+y=0\\–x+2y=–5\) because \(2(1)+(–2)=0\\–1+2(–2)=–5\) makes both equations true.
One way to solve a system of linear equations is by substitution.
STEP BY STEP
Step 1. Solve one equation for one of the variables.
Step 2. Substitute the expression from Step 1 into the other equation and solve for the other variable.
Step 3. Substitute the value from Step 2 into one of the original equations and solve.
All systems of equations can be solved by substitution for any one of the four variables in the problem. The most efficient way of solving is locating the 1x or 1y in the equations because this eliminates the possibility of having fractions in the equations.
Examples
Solving a System of Equations by Substitution Review
Solving a System of Equations by Elimination
Another way to solve a system of linear equations is by elimination.
STEP BY STEP
Step 1. Multiply, if necessary, one or both equations by a constant so at least one pair of like terms has opposite coefficients.
Step 2. Add the equations to eliminate one of the variables.
Step 3. Solve the resulting equation.
Step 4. Substitute the value from Step 3 into one of the original equations and solve for the other variable.
All system of equations can be solved by the elimination method for any one of the four variables in the problem. One way of solving is locating the variables with opposite coefficients and adding the equations. Another approach is multiplying one equation to obtain opposite coefficients for the variables.
Examples
Solving a System of Equations by Elimination Review
Solving a System of Equations by Graphing
Graphing is a third method of a solving system of equations. The point of intersection is the solution for the graph. This method is a great way to visualize each graph on a coordinate plane.
STEP BY STEP
Step 1. Graph each equation in the coordinate plane.
Step 2. Estimate the point of intersection.
Step 3. Check the point by substituting for x and y in each equation of the original system.
The best approach to graphing is to obtain each line in slope-intercept form. Then, graph the y-intercept and use the slope to find additional points on the line.
Example
Solving a System of Equations by Graphing Review
Solving a System of a Linear Equation and an Equation of a Circle
There are many other types of systems of equations. One example is the equation of a line y = mx and the equation of a circle \(x^2 + y^2 = r^2\) where r is the radius. With this system of equations, there can be two ordered pairs that intersect between the line and the circle. If there is one ordered pair, the line is tangent to the circle.
KEEP IN MIND
There will be two solutions in many cases with the system of a linear equation and an equation of a circle.
This system of equations is solved by substituting the expression mx in for y in the equation of a circle. Then, solve the equation for x. The values for x are substituted into the linear equation to find the value for y.
Example
Solving a System of a Linear Equation and the Equation of a Circle Review
Let’s Review!
- There are three ways to solve a system of equations: graphing, substitution, and elimination. Using any method will result in the same solution for the system of equations.
- Solving a system of a linear equation and an equation of a circle uses substitution and usually results in two solutions.
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