Math Problem Statement
Solve the system of equations below:
-5x + 3y + z = 5
-3x - 3y - z = 3
-x + 5z = 1
Solution
The system of equations provided in the image is:
-5x + 3y + z &= 5 \\ -3x - 3y - z &= 3 \\ -x + 5z &= 1 \end{align*}$$ We can solve this system using substitution, elimination, or matrix methods. I'll proceed by using substitution and elimination to find the values of $$x$$, $$y$$, and $$z$$. ### Step 1: Simplify the third equation The third equation is: $$-x + 5z = 1$$ Solving for $$x$$: $$x = 5z - 1$$ ### Step 2: Substitute $$x = 5z - 1$$ into the first and second equations Substituting into the first equation: $$-5(5z - 1) + 3y + z = 5$$ $$-25z + 5 + 3y + z = 5$$ $$-24z + 3y = 0 \quad \text{(Equation A)}$$ Now, substitute into the second equation: $$-3(5z - 1) - 3y - z = 3$$ $$-15z + 3 - 3y - z = 3$$ $$-16z - 3y = 0 \quad \text{(Equation B)}$$ ### Step 3: Solve the system of equations from Step 2 We now have two equations in terms of $$z$$ and $$y$$: 1. $$-24z + 3y = 0$$ 2. $$-16z - 3y = 0$$ Add these two equations: $$-24z + 3y + (-16z - 3y) = 0 + 0$$ $$-40z = 0$$ $$z = 0$$ ### Step 4: Solve for $$y$$ and $$x$$ Substitute $$z = 0$$ into one of the simplified equations. Using Equation A: $$-24(0) + 3y = 0$$ $$3y = 0$$ $$y = 0$$ Now, substitute $$z = 0$$ into the expression for $$x$$: $$x = 5(0) - 1$$ $$x = -1$$ ### Final Solution: $$x = -1, \quad y = 0, \quad z = 0$$ Would you like more details or have any questions? --- Here are 5 related questions: 1. How can the substitution method be applied to systems with more than three variables? 2. What is the elimination method and when is it preferable to use it? 3. How can matrix methods (e.g., Gaussian elimination) solve systems of equations efficiently? 4. What are the conditions for a system of equations to have a unique solution? 5. How can we verify if a system of equations is consistent or inconsistent? **Tip:** Always check your solution by substituting the values of the variables back into the original equations.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Formulas
Substitution Method
Elimination Method
Solving Linear Equations
Theorems
Properties of Equality
Solution of Linear Systems
Suitable Grade Level
Grades 9-12
Related Recommendation
Solving a System of Three Linear Equations Involving Variables x, y, and z
Solving a System of Linear Equations for x, y, and z
Solving Systems of Linear Equations Using Elimination and Substitution
Solve the System of Linear Equations 3x + y + z = 15, 2x - 3y + z = 0, x - y - 2z = -19
Solving a System of Linear Equations: 55x + 44y - 55z = -36, 22x - 44y + 22z = 44, 66x - 33y + 66z = -24