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System of Linear Equations in 2 variablesHow to Solve Systems What is a system of equations?AnswerA system of equation just means 'more than 1 equation.'. A system of linear equations is just more than 1 line, see the picture: Ok, so what is the solution of a system of equations?AnswerThe solution is where the equations 'meet' or intersect. The red point is the solution of the system. How many solutions can systems of linear equations have?AnswerThere can be zero solutions, 1 solution or infinite solutions--each case is explained in detail below. Note: Although systems of linear equations can have 3 or more equations,we are going to refer to the most common case--a stem with exactly 2 lines. Case I: 1 Solution This is the most common situation and it involves lines that intersect at exactly one point. Case 2: No Solutions This only happens when the lines are parallel. As you can see, parallel lines are not going to ever meet. Example of a stem that has no solution:
Case III: Infinite Solutions This is the rarest case and only occurs when you have the same line Example of a system that has infinite solutions:
How can we find solutions to systems of equations?To find the solution to systems of linear equations, you can any of the methods below: Video on Solutions of Systems of Equations
Walk through our printable solving systems of equations worksheets to learn the ins and outs of solving a set of linear equations. Ensure students are thoroughly informed of the methods of elimination, substitution, matrix, cross-multiplication, Cramer's Rule, and graphing that are crucial for arriving at the solutions. Reaffirm skills in graphing the equation of a line as a prerequisite to solve the pairs of simultaneous equations by graphing them. This array of free pdf worksheets is meticulously designed for 8th grade and high school students. CCSS: HSA-REI.6 A linear equation in two variables is an equation of the form ax + by + c = 0 where a, b, c ∈ R, a, and b ≠ 0. When we consider a system of linear equations, we can find the number of solutions by comparing the coefficients of the variables of the equations. Also, we can find the number of solutions by the graphical method. In this article, we will learn how to determine the number of solutions in a system of equations with two variables. Three Types of Solutions of a System of Linear EquationsConsider the pair of linear equations in two variables x and y. a1x + b1y + c1 = 0 a2x + b2y + c2 = 0 Here a1, b1, c1, a2, b2, c2 are all real numbers. Note that, a12 + b12 ≠ 0, a22 + b22 ≠ 0 1. If (a1/a2) ≠ (b1/b2), then there will be a unique solution. If we plot the graph, the lines will intersect. This type of equation is called a consistent pair of linear equations. 2. If (a1/a2) = (b1/b2) = (c1/c2), then there will be infinitely many solutions. The lines will coincide. This type of equation is called a dependent pair of linear equations in two variables 3. If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution. If we plot the graph, the lines will be parallel. This type of equation is called an inconsistent pair of linear equations. In Short:
ExampleHow many solutions does the following system have? y = -2x – 4 y = 3x + 3 Solution: Given y = -2x – 4 y = 3x + 3 Rewriting to the general form -2x – y – 4 = 0 3x – y + 3 = 0 Comparing the coefficients, (a1/a2) = -⅔ (b1/b2) = -1/-1 = 1 (a1/a2) ≠ (b1/b2) Hence, this system of equations will have only one solution. Frequently Asked QuestionsThe general form of a
linear equation in two variables is given by ax + by + c = 0, where x and y are the variables. If we plot the graph of an inconsistent pair of linear equations, the lines will be parallel. Suppose we plot the graph of a pair of linear equations. If both the lines intersect at a point, there exists a unique solution. This type of linear equations are called consistent pair of linear equations. |
Number of solutions to a system of equations worksheet
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