Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 34
\(29-44\) Use Cramer's Rule to solve the system. $$ \left\\{\begin{array}{l}{10 x-17 y=21} \\ {20 x-31 y=39}\end{array}\right. $$
Short Answer
Expert verified
x = \(\frac{2}{5}\), y = -1.
Step by step solution
01
Identify the coefficients
The given system of linear equations is \(10x - 17y = 21\) and \(20x - 31y = 39\). Identify the coefficients: \(a_1 = 10\), \(b_1 = -17\), \(c_1 = 21\); \(a_2 = 20\), \(b_2 = -31\), \(c_2 = 39\).
02
Calculate the determinant of the coefficient matrix
The determinant \(D\) of the coefficient matrix is calculated as: \[D = \begin{vmatrix} 10 & -17 \ 20 & -31 \end{vmatrix} = (10)(-31) - (-17)(20)\].
03
Simplify the determinant
Calculate the components: \((10)(-31) = -310\) and \((-17)(20) = -340\). Therefore, the determinant \(D = -310 + 340 = 30\).
04
Calculate the determinant for x (D鈧�)
Replace the x-coefficients with the constants to find \(D_{x}\):\[D_{x} = \begin{vmatrix} 21 & -17 \ 39 & -31 \end{vmatrix} = (21)(-31) - (-17)(39)\].
05
Simplify D鈧�
Calculate the components: \((21)(-31) = -651\) and \((-17)(39) = -663\). Therefore, \(D_{x} = -651 + 663 = 12\).
06
Calculate the determinant for y (D岬�)
Replace the y-coefficients with the constants to find \(D_{y}\):\[D_{y} = \begin{vmatrix} 10 & 21 \ 20 & 39 \end{vmatrix} = (10)(39) - (21)(20)\].
07
Simplify D岬�
Calculate the components: \((10)(39) = 390\) and \((21)(20) = 420\). Therefore, \(D_{y} = 390 - 420 = -30\).
08
Solve for x and y
Using Cramer's Rule, x and y can be calculated as follows:\[ x = \frac{D_{x}}{D} = \frac{12}{30} = \frac{2}{5} \]\[ y = \frac{D_{y}}{D} = \frac{-30}{30} = -1 \]
09
Verify the solution
Substitute \(x = \frac{2}{5}\) and \(y = -1\) back into the original equations to verify:For \(10x - 17y = 21\): \(10\left(\frac{2}{5}\right) + 17 = 21\). For \(20x - 31y = 39\): \(20\left(\frac{2}{5}\right) + 31 = 39\). Both checks confirm the solution.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Determinant
A determinant is a special number that can be computed from a square matrix. In the context of solving a system of linear equations using Cramer's Rule, the determinant plays a crucial role. It helps us evaluate whether a unique solution exists.
For a 2x2 matrix:
For a 2x2 matrix:
- The determinant is calculated using the formula: \(D = ad - bc\), where \(a, b, c,\) and \(d\) are the elements of the matrix.
- If the determinant is zero, the system does not have a unique solution.
- Otherwise, the determinant provides information that leads us to the solution using Cramer鈥檚 Rule.
System of Linear Equations
A system of linear equations involves finding values for variables that satisfy all equations within the system. Each equation in the system represents a line in a 2D space. In our example, the system given was:
In our problem, we used Cramer's Rule to find this solution. Cramer鈥檚 Rule can only be applied if:
- \(10x - 17y = 21\)
- \(20x - 31y = 39\)
In our problem, we used Cramer's Rule to find this solution. Cramer鈥檚 Rule can only be applied if:
- The system is square, meaning the number of equations matches the number of variables.
- The determinant of the coefficient matrix is non-zero to ensure a unique solution.
Matrix Algebra
Matrix algebra is a vital mathematical tool for handling systems of equations. In matrix form, our system of equations is represented by aligning coefficients into a matrix:The coefficients from our system:
- Matrix \(A = \begin{pmatrix} 10 & -17 \ 20 & -31 \end{pmatrix}\)
- Constant matrix \(C = \begin{pmatrix} 21 \ 39 \end{pmatrix}\)
- The determinant of the coefficient matrix helps check the existence of a unique solution.
- Replacements of columns in the matrix are done to solve for variables \(x\) and \(y\), using Cramer's Rule.
