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Solve each \(3\)X\(3\) system of linear equations.
\(\textbf{1)}\) \(2x+3y-5z=-7\)
\(\hspace{12pt}3x-6y+4z=3 \)
\(\hspace{12pt}x+4y+2z=15 \)
\(\textbf{2)}\) \(4x-2y+3z=20\)
\(\hspace{12pt}1x-2y+3z=14\)
\(\hspace{12pt}4x+5y-8z=-24\)
\(\textbf{3)}\) \( x+y=6 \)
\(\hspace{12pt}y+z=10 \)
\(\hspace{12pt}x+z=8 \)
\(\textbf{4)}\) \(10x+y+10z=42\)
\(\hspace{12pt}11x+20y+4z=84\)
\(\hspace{12pt}x+40y+z=84\)
\(\textbf{5)}\) \(3x+y-2z=10\)
\(\hspace{12pt}x+y+z=9\)
\(\hspace{12pt}x+3y+z=11\)
\(\textbf{6)}\) \( x+y=0 \)
\(\hspace{12pt}y+z=-4 \)
\(\hspace{12pt}x+z=6 \)
\(\textbf{7)}\) \(x-2y+3z=4\)
\(\hspace{12pt}2x+y-4z=3\)
\(\hspace{12pt}-3x+4y-z=-2\)
\(\textbf{8)}\) \(2x+3y+z=11\)
\(\hspace{12pt}x+4y+2z=7\)
\(\hspace{12pt}3x+2y+5z=20\)
\(\textbf{9)}\) The sum of three numbers is \(19\). The 1st number is double the second number. The difference between the 2nd number and 3rd number is \(7\).
\(\textbf{10)}\) The sum of three numbers is \(10\). The 1st number is \(14\) less than the second. The 2nd number is \(3\) less than the 3rd number.
Related Pages\(\)
\(\bullet\text{ Algebra 2/Precalc Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ 3 Equation System Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(wolframalpha.com)}\)
\(\bullet\text{ Solving Systems with Substitution}\)
\(\,\,\,\,\,\,\,\,y=−2x+5\)
\(\,\,\,\,\,\,\,\,y=x-1…\)
\(\bullet\text{ Solving Systems with Elimination}\)
\(\,\,\,\,\,\,\,\,7x+4y=31 \)
\(\,\,\,\,\,\,\,\,3x+2y=15…\)
\(\bullet\text{ Graphing Systems of Inequalities}\)
\(\,\,\,\,\,\,\,\,y\lt-3x+2 \)
\(\,\,\,\,\,\,\,\,y\ge\frac{1}{2}x-1…\)
\(\bullet\text{ Nonlinear Systems}\)
\(\,\,\,\,\,\,\,\,x^2+y^2=8 \)
\(\,\,\,\,\,\,\,\,y=x…\)
\(\bullet\text{ Andymath Homepage}\)
In Summary
Solving 3 variable systems involves finding the values of the three variables that simultaneously satisfy a system of equations with three variables. This can be done using various methods, such as substitution, elimination, and graphing. It can be done with equations by hand or as a shortcut you can use matrices on a calculator.
Solving 3 variable systems is typically covered in a high school or college algebra class. Learning about three-variable systems is important because they are used to model and analyze a wide range of real-world situations. These systems can be used to solve problems in fields such as economics, physics, and engineering. Understanding how to work with three-variable systems can help students develop critical thinking and problem-solving skills that are useful in many different fields.
Mistakes commonly occur when substituting values into the equations. It’s important to be careful and methodical when working with three-variable systems to avoid these types of mistakes. Practice definitely reduces that chances of these mistakes.
Topics related to 3 Variable Systems:
Linear programming: This is a mathematical method used to maximize or minimize a linear objective function subject to a set of linear constraints. Linear programming problems often involve finding the optimal solution to a system of linear equations that have three or more variables.
Matrix operations: Matrices are often used to represent and manipulate systems of equations with three or more variables. Matrix operations, such as matrix multiplication and inversion, can be used to solve systems of linear equations and perform other mathematical tasks.
Vector calculus: Vector calculus is a branch of mathematics that deals with the study of vector fields, which are functions that associate a vector with every point in space. Vector calculus involves the use of 3-variable systems to describe the behavior of vector fields, such as the flow of a fluid or the movement of a particle.
Differential equations: A differential equation is an equation that relates a function and its derivatives. Solving differential equations often involves finding the values of variables that satisfy the equation, which may involve 3 or more variables.
Multivariate statistics: Multivariate statistics is a branch of statistics that deals with the analysis of data sets with multiple variables. In multivariate statistics, 3-variable systems are often used to model the relationships between variables and to make predictions about future data.
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