Math Problem Statement
Solution
Let's go through each equation in the image and solve them step-by-step.
Solution:
1.
2.
4 - 6y - y = 1 \\ -7y = -3 \\ y = \frac{3}{7}$$ Using $$y$$ in the first equation to find $$x$$: $$x + 3\left(\frac{3}{7}\right) = 2 \\ x + \frac{9}{7} = 2 \\ x = 2 - \frac{9}{7} \\ x = \frac{14}{7} - \frac{9}{7} \\ x = \frac{5}{7}$$ So, $$x = \frac{5}{7}$$ and $$y = \frac{3}{7}$$. #### 3. $$3x + 2y = 10$$ Using substitution for $$x$$: $$3\left(\frac{5}{7}\right) + 2y = 10 \\ \frac{15}{7} + 2y = 10 \\ 2y = 10 - \frac{15}{7} \\ 2y = \frac{70}{7} - \frac{15}{7} \\ 2y = \frac{55}{7} \\ y = \frac{55}{14}$$ #### 4. $$2y - x = 3$$ Substitute $$y$$: $$2\left(\frac{55}{14}\right) - \frac{5}{7} = 3 \\ \frac{110}{14} - \frac{10}{14} = 3 \\ \frac{100}{14} = 3 \\ y = \frac{15}{2}$$ #### 5. $$2x - y + z = 3$$ Using substitution for $$x$$ and $$y$$: $$2\left(\frac{5}{7}\right) - \frac{15}{2} + z = 3 \\ \frac{10}{7} - \frac{105}{14} + z = 3 \\ z = 3 + \frac{105}{14} - \frac{10}{7}$$ #### 6. $$x + 3y + z = 6$$ Using $$x$$ and $$y$$: $$\frac{5}{7} + 3\left(\frac{15}{2}\right) + z = 6 \\ z = 6 - \frac{45}{2} - \frac{5}{7}$$ #### 7. $$y - 2z = 1$$ Substituting $$y$$: $$\frac{15}{2} - 2z = 1 \\ z = \frac{7}{2}$$ #### 8. $$3x + 2y - z = 7$$ Substituting $$x$$ and $$y$$: $$3\left(\frac{5}{7}\right) + 2\left(\frac{15}{2}\right) - \frac{7}{2} = 7$$ ### Summary: This initial step-by-step attempt shows a consistent method of solving for $$x$$, $$y$$, and $$z$$, although intermediate steps need verification. Would you like details or have any questions about these steps? Here are 8 related questions: 1. How can we verify the solutions to these linear equations? 2. What methods are available for solving systems of equations? 3. Can these solutions be represented graphically? 4. How do we solve systems with three variables? 5. What are the properties of consistent, inconsistent, and dependent systems? 6. How can matrices be used to solve these systems? 7. What is the role of determinant in solving linear systems? 8. How can substitution and elimination methods be compared? Tip: Always check your solutions by substituting back into the original equations to ensure accuracy.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear equations
Systems of equations
Substitution method
Elimination method
Formulas
-
Theorems
-
Suitable Grade Level
High school
Related Recommendation
Solving Systems of Linear Equations Using Elimination and Substitution Methods
Solving a System of Linear Equations: x - 4y + 4z = -1, -x - z + 3y = 5, 2x + 5z - 6 = y
Solving a System of Linear Equations with Three Variables
Solving System of Equations: x + y + z = 3, x - y + z = 2, x + y - 2z = 0
Solve a 3-Variable System of Linear Equations Using Elimination and Substitution