Math Problem Statement

Find the values of x, y, and z given the equations: 2x + y + 3z = 3, x + 5z = -2, and z = -1.

Solution

The question appears to ask for the values of x,y,x, y, and zz based on a set of linear equations. Let me extract the details from the image to analyze and solve the equations.

Here are the equations extracted:

  1. 2x+y+3z=32x + y + 3z = 3
  2. x+5z=2x + 5z = -2
  3. z=1z = -1

Now, let's solve for x,y,x, y, and zz:

Step 1: Substitute z=1z = -1 into the other equations

From Equation (3), z=1z = -1.

Substitute z=1z = -1 into Equation (2):

x+5(1)=2x + 5(-1) = -2 x5=2x=3x - 5 = -2 \quad \Rightarrow \quad x = 3

Substitute z=1z = -1 and x=3x = 3 into Equation (1):

2(3)+y+3(1)=32(3) + y + 3(-1) = 3 6+y3=3y=06 + y - 3 = 3 \quad \Rightarrow \quad y = 0

Final Solution:

x=3,  y=0,  z=1x = 3, \; y = 0, \; z = -1

Would you like me to verify these values or explain the steps in more detail?


Here are 5 related questions for further practice:

  1. What is the determinant of the coefficient matrix of these equations?
  2. How would the solution change if z1z \neq -1?
  3. Can you graphically interpret the solution of these linear equations?
  4. How would you use matrix methods to solve these equations?
  5. What if one equation is removed—how would you find the solutions in terms of parameters?

Tip: Always substitute back the variables to verify your solution in all equations.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Substitution Method
Algebra

Formulas

Substitution of variables into equations
Linear equation solving techniques

Theorems

Properties of equality in algebra

Suitable Grade Level

Grades 7-9