Solving a System of Linear Equations to Find b

We are given a system of two linear equations with two variables, 'a' and 'b'. We need to find the value of 'b'.

The given equations are:

1. Equation 1: \(a + 2b = 55\)
2. Equation 2: \(a - 2b = -13\)

We can solve this system using either the substitution method or the elimination method. The elimination method is particularly suitable here because the coefficients of 'b' are equal in magnitude but opposite in sign (+2b and -2b).

Using the Elimination Method

To eliminate the variable 'b', we can add Equation 1 and Equation 2.

Adding the left-hand sides (LHS) and the right-hand sides (RHS):

\((a + 2b) + (a - 2b) = 55 + (-13)\)

Combine like terms on the LHS:

\(a + a + 2b - 2b = 55 - 13\)

\(2a + 0b = 42\)

\(2a = 42\)

Now, solve for 'a' by dividing both sides by 2:

\(a = \frac{42}{2}\)

\(a = 21\)

Substituting the Value of a to Find b

Now that we have the value of 'a', we can substitute it into either Equation 1 or Equation 2 to solve for 'b'. Let's use Equation 1:

Substitute \(a = 21\) into Equation 1:

\(21 + 2b = 55\)

Subtract 21 from both sides of the equation:

\(2b = 55 - 21\)

\(2b = 34\)

Now, solve for 'b' by dividing both sides by 2:

\(b = \frac{34}{2}\)

\(b = 17\)

Verification

To check our solution, we can substitute \(a = 21\) and \(b = 17\) into the second equation, Equation 2:

\(a - 2b = -13\)

\(21 - 2(17) = -13\)

\(21 - 34 = -13\)

\(-13 = -13\)

Since both sides are equal, our values for 'a' and 'b' are correct.

The value of 'b' is 17.

Comparing with Options

The calculated value of 'b' is 17, which corresponds to option C.

Revision Table: System of Equations Methods

Additional Information: Solving Systems of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same variables. The solution to a system of two linear equations in two variables (\(x\) and \(y\)) is an ordered pair \((x, y)\) that satisfies both equations simultaneously. Graphically, this solution corresponds to the point of intersection of the lines represented by the equations.

There are several methods to solve a system of linear equations:

• Graphing Method: Plot both equations on the same coordinate plane. The point where the lines intersect is the solution. This method can be less precise if the intersection point does not have integer coordinates.
• Substitution Method:
  1. Solve one of the equations for one variable in terms of the other variable.
  2. Substitute this expression into the other equation. This results in a single equation with one variable.
  3. Solve the resulting equation for the single variable.
  4. Substitute the value found back into the expression from step 1 to find the value of the other variable.
• Elimination Method (or Addition Method):
  1. Multiply one or both equations by a constant so that the coefficients of one variable are opposites (e.g., 3x and -3x) or the same (e.g., 3x and 3x).
  2. Add or subtract the equations to eliminate one variable.
  3. Solve the resulting equation for the single variable.
  4. Substitute the value found back into one of the original equations to find the value of the other variable.

The given problem was efficiently solved using the elimination method by adding the two equations.