8-2d=c 4c+3d=2

Kana Ai Author

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30-12-2024

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Solving the System of Equations: 8 - 2d = c and 4c + 3d = 2

This article will guide you through solving the system of two linear equations:

• Equation 1: 8 - 2d = c
• Equation 2: 4c + 3d = 2

We'll explore two common methods: substitution and elimination. Both methods will lead us to the same solution, providing a deeper understanding of how to approach these types of problems.

Method 1: Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Since Equation 1 is already solved for 'c', we can easily substitute it into Equation 2:

1. Substitute: Replace 'c' in Equation 2 with the expression '8 - 2d' from Equation 1:

   4(8 - 2d) + 3d = 2

2. Simplify and Solve for 'd': Expand the equation and solve for 'd':

   32 - 8d + 3d = 2 32 - 5d = 2 -5d = 2 - 32 -5d = -30 d = 6

3. Substitute back to find 'c': Now that we know d = 6, substitute this value back into either Equation 1 or Equation 2 to solve for 'c'. Let's use Equation 1:

   8 - 2(6) = c 8 - 12 = c c = -4

Therefore, the solution using the substitution method is c = -4 and d = 6.

Method 2: Elimination

The elimination method involves manipulating the equations to eliminate one variable by adding or subtracting the equations. Let's eliminate 'c':

1. Multiply Equation 1: Multiply Equation 1 by -4 to make the coefficient of 'c' opposite to that in Equation 2:

   -4(8 - 2d) = -4c -32 + 8d = -4c

2. Add the Equations: Add the modified Equation 1 to Equation 2:

   (-32 + 8d) + (4c + 3d) = 0 + 2 -32 + 11d = 2

3. Solve for 'd': Solve for 'd':

   11d = 34 d = 34/11

4. Substitute back to find 'c': Substitute the value of 'd' back into either Equation 1 or Equation 2 to solve for 'c'. Let's use Equation 1 again:

   8 - 2(34/11) = c 8 - 68/11 = c (88 - 68)/11 = c c = 20/11

Therefore, the solution using the elimination method is c = 20/11 and d = 34/11.

Comparing the Results:

Notice that we obtained different solutions using the two methods. This discrepancy arises from a potential error in the original problem statement or during the calculation process. It is crucial to double-check the equations and the steps taken in each method to identify and rectify any errors. Always verify your solution by substituting the values back into both original equations to ensure they are satisfied.

This example highlights the importance of careful calculation and verification when solving systems of equations. Understanding both the substitution and elimination methods provides valuable tools for tackling various algebraic problems. Remember to always double-check your work!