How to Solve Linear Systems of Equations by Graphing
Concept
In mathematics, a system of equations, also known as a set of simultaneous or an equation system, is a finite set of equations for which we sought the common solutions. A system of equations can be classified in a similar manner as single equations.
Any system of equations can be solved in different methods. To solve a system of equations in 2 variables, we need at least 2 equations. Similarly, for solving a system of equations in 3 variables, we will require at least 3 equations. Let us understand 3 ways to solve a system of equations given the equations are linear equations in two variables.
Substitution Method
Elimination Method
Graphical Method
In the graphical method, the solution of simultaneous equations is obtained by plotting their graphs. The point of intersection of the two lines is the solution of the system of equations using graphical method.
Rules
1. Make sure both equations are in slope-intercept form. If any or both equations are in standard form, convert them into slope-intercept form.
2. Graph both lines using y-intercept and slopes then find the point of intersection.
3. The point of intersection will be the point where the two lines intersect.
4. If the lines are parallel then there is no solution as parallel lines can never intersect.
5. If the lines lie on top of each other, there are infinite solutions.
Example
What is the solution to the following system?
Solution
1. Change the first equation into slope-intercept form.
2. Graph both lines using y-intercept and slopes then find the point of intersection.
3. The point of intersection will be the point where the two lines intersect.
The solution to the system is (0, 3)
Practice How to Solve Linear Systems of Equations by Graphing
Practice Problem 1
What is the solution to the following system?
Practice Problem 2
What is the solution to the following system?
Practice Problem 3
Betty graphed a line using the equation . She graphs another line that yields infinite solutions. Which line could be the second line that she graphed?
A system of linear equations – is a set of equations which are satisfied by the same set of variables.
Coordinate Point – A pair of numbers used to locate a point in the coordinate plane.
Coordinate Plane – A coordinate system in which a horizontal number line and a vertical number line intersect at their zero points.
Intersect – To cross each other.
Variable – A symbol, usually a letter, used to represent a number in mathematical expressions or sentences.
Equation – A mathematical sentence stating that two quantities are equal.
Slope intercept form – a linear equation written in the form y = mx + b, where m stands for the slope, and b stands for the y-intercept.
Standard form – a linear equation written in the form Ax + By = C
Solution – A solution to a system of equations is the value of the variables that makes ALL the equations True.
Pre-requisite Skills
Solve Equations with Rational Coefficients
Solve Two-Step Equations
Write Two-Step Equations
Solve Two-Sided Equations with Rational Coefficients
Solve Multi-Step Equations
Slope-intercept Form of an Equation
Graphing Lines Using Intercepts
Write Linear Equations
Direct Variation Equations (y = mx)
Solutions of Systems of Equations