Mathematics
Grade 10
15 min
Solve a system of equations using augmented matrices word problems
Solve a system of equations using augmented matrices word problems
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1
Introduction & Learning Objectives
Learning Objectives
Translate a real-world word problem into a system of linear equations.
Construct an augmented matrix from a system of linear equations.
Correctly apply elementary row operations to an augmented matrix.
Transform an augmented matrix into row-echelon form to find a solution.
Interpret the solution from a final matrix in the context of the original word problem.
Verify a solution by substituting the values back into the original equations.
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This tutorial will teach you how to take a word problem, turn it into a system of equations, and organize it into a special grid...
2
Key Concepts & Vocabulary
TermDefinitionExample
System of Linear EquationsA set of two or more linear equations that share the same variables. The solution to the system is the set of values for the variables that makes all equations true simultaneously.The equations 2x + y = 7 and x - y = 2 form a system. The solution is x=3, y=1.
Augmented MatrixA matrix that represents a system of linear equations. It consists of the coefficients of the variables and the constant terms, separated by a vertical line.For the system 2x + y = 7 and x - y = 2, the augmented matrix is [ 2 1 | 7 ]
[ 1 -1 | 2 ]
CoefficientA numerical or constant quantity placed before and multiplying the variable in an algebraic expression.In the term 5x, the coefficient is 5.
Elementary Row OperationsA set of three specific operations that can be per...
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Core Formulas
Row Swap
R_i \leftrightarrow R_j
Swap the position of row 'i' with row 'j'. This is useful for getting a '1' into the top-left position if it's available in another row.
Scalar Multiplication
kR_i \rightarrow R_i
Multiply every element in row 'i' by a non-zero constant 'k'. This is used to create a leading '1' in a row.
Row Addition
R_i + kR_j \rightarrow R_i
Add a multiple ('k') of row 'j' to row 'i' and replace row 'i' with the result. This is the key operation used to create zeros below the leading '1's.
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Challenging
A farmer is planting a 30-acre field with corn (c), soybeans (s), and wheat (w). The cost is $50/acre for corn, $40/acre for soybeans, and $30/acre for wheat, with a total budget of $1220. The farmer wants to plant twice as many acres of corn as wheat. How many acres of each crop should be planted?
A.c=6, s=21, w=3
B.c=4, s=24, w=2
C.c=8, s=18, w=4
D.c=2, s=26, w=1
Challenging
A school is selling T-shirts (t), hats (h), and sweatshirts (s) for a fundraiser. They have the following information: 1) The total number of items sold is 100. 2) The total revenue is $1500. 3) The number of T-shirts sold is equal to the number of hats and sweatshirts combined. When you set up and reduce the augmented matrix, you get the row [0 1 1 | 50]. What can you conclude?
A.The problem has no unique solution; you can only determine the combined number of hats and sweatshirts.
B.There is no solution because the information is contradictory.
C.Exactly 50 hats and 50 sweatshirts were sold.
D.An error was made, as a unique solution should be possible.
Challenging
A mixture of nuts contains almonds (a), walnuts (w), and cashews (c). The total weight is 10 kg. The cost is $15/kg for almonds, $12/kg for walnuts, and $18/kg for cashews, for a total cost of $147. The ratio of the weight of walnuts to almonds is 2:1. How many kilograms of cashews are in the mixture?
A.2 kg
B.3 kg
C.4 kg
D.5 kg
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