Math Problem Statement

a+b+c=0 4a+2b+c=-4 9a+3b+c=-18

Solution

We have a system of three linear equations:

  1. a+b+c=0a + b + c = 0
  2. 4a+2b+c=44a + 2b + c = -4
  3. 9a+3b+c=189a + 3b + c = -18

Step 1: Eliminate cc

To eliminate cc, we will subtract the first equation from the second and the second from the third.

Subtract equation (1) from equation (2):

(4a+2b+c)(a+b+c)=40(4a + 2b + c) - (a + b + c) = -4 - 0 3a+b=43a + b = -4 This is equation (4).

Subtract equation (2) from equation (3):

(9a+3b+c)(4a+2b+c)=18(4)(9a + 3b + c) - (4a + 2b + c) = -18 - (-4) 5a+b=145a + b = -14 This is equation (5).

Step 2: Solve the system of equations with two variables

Now we have two equations:

  1. 3a+b=43a + b = -4
  2. 5a+b=145a + b = -14

Subtract equation (1) from equation (2): (5a+b)(3a+b)=14(4)(5a + b) - (3a + b) = -14 - (-4) 2a=102a = -10 a=5a = -5

Step 3: Substitute a=5a = -5 into one of the equations

Substitute a=5a = -5 into equation 3a+b=43a + b = -4: 3(5)+b=43(-5) + b = -4 15+b=4-15 + b = -4 b=11b = 11

Step 4: Solve for cc

Now substitute a=5a = -5 and b=11b = 11 into the first equation a+b+c=0a + b + c = 0: 5+11+c=0-5 + 11 + c = 0 6+c=06 + c = 0 c=6c = -6

Final Solution:

a=5,b=11,c=6a = -5, \quad b = 11, \quad c = -6

Let me know if you want any details or have questions.

Here are 5 related questions:

  1. How can you verify if the solution is correct by substituting it back into the original equations?
  2. What is the general method for solving systems of equations using elimination?
  3. How would you solve this system using matrix methods such as Gaussian elimination?
  4. Can this system be solved using substitution instead of elimination?
  5. How would this problem change if the equations were non-linear?

Tip: To quickly check your solution, always substitute the values back into the original equations to confirm they hold true.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

Equation Elimination Method
Substitution Method

Theorems

Elimination of Variables
Solving Systems of Linear Equations

Suitable Grade Level

Grades 9-12