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Name: NOTES 10: SYSTEMS OF EQUATIONS AND INEQUALITIES Date: Period: Mrs. Nguyen's Initial: LESSON 10.1 SYSTEMS OF EQUATIONS Definition of a system of equations A system of equations is a set of 2
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How to fill out ystems of equations and
How to fill out systems of equations and:
01
Understand the concept of systems of equations: A system of equations is a set of two or more equations containing the same variables. The goal is to find the values of the variables that satisfy all the equations simultaneously.
02
Identify the variables: Look at the equations and identify the variables involved. For example, if the system involves two equations:
Equation 1: 2x + 3y = 10
Equation 2: 4x - y = 5
The variables in this case are x and y.
Choose a method for solving the system: There are several methods to solve systems of equations, such as substitution, elimination, or graphing. Choose the method that suits your preference or the requirements of your problem.
Solve the system using the chosen method:
Substitution Method: Solve one equation for one variable and substitute the result into the other equation. For example:
Equation 1: y = 10 - 2x
Substitute the value of y in Equation 1 into Equation 2:
4x - (10 - 2x) = 5
Simplify and solve for x:
4x - 10 + 2x = 5
6x - 10 = 5
6x = 15
x = 2.5
Substitute the value of x in Equation 1 to find y:
y = 10 - 2(2.5)
y = 10 - 5
y = 5
The solution to the system is x = 2.5 and y = 5.
Elimination Method: Multiply one or both equations by appropriate constants to make the coefficients of one variable in both equations equal. Then add or subtract the equations to eliminate one variable. For example:
Equation 1: 2x + 3y = 10
Equation 2: 4x - y = 5
Multiply Equation 1 by 2 and Equation 2 by 3 to make the coefficients of x equal:
4x + 6y = 20
12x - 3y = 15
Add the equations to eliminate y:
16x = 35
x = 2.1875
Substitute the value of x in Equation 1 to find y:
2(2.1875) + 3y = 10
375 + 3y = 10
3y = 5.625
y = 1.875
The solution to the system is x = 2.1875 and y = 1.875.
Check the solution: Substitute the values of x and y into each equation to ensure they satisfy all the equations. If the solution satisfies all the equations, then it is the correct solution to the system.
Who needs systems of equations and:
01
Students studying algebra: Systems of equations are fundamental in algebra courses. Learning how to solve systems of equations helps students develop problem-solving and critical thinking skills.
02
Engineers and scientists: Systems of equations are used in various fields to model and solve real-life problems. Engineers use systems of equations to analyze circuits, optimize processes, and solve structural problems. Scientists utilize systems of equations to model physical phenomena, such as population growth or chemical reactions.
03
Economists and business analysts: Systems of equations are used to model economic systems and market behavior. Economists analyze supply and demand relationships, market equilibrium, and economic policies using systems of equations. Business analysts utilize systems of equations for forecasting and analyzing business trends.
In conclusion, understanding how to fill out systems of equations and is essential for students studying algebra, as well as professionals in engineering, science, economics, and business.
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What is systems of equations and?
Systems of equations are a set of multiple equations that are solved simultaneously to find the common solution.
Who is required to file systems of equations and?
Individuals or organizations dealing with multiple unknown variables in mathematical problems are required to file systems of equations.
How to fill out systems of equations and?
Systems of equations can be filled out by writing down each equation, identifying the variables, and solving for their values using various methods such as substitution or elimination.
What is the purpose of systems of equations and?
The purpose of systems of equations is to find the common solution that satisfies all the equations in the set.
What information must be reported on systems of equations and?
The information reported on systems of equations includes the equations themselves, the variables involved, and the solutions obtained.
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