Question

Suppose \[A\] and \[B\] represent linear expressions. If \[A + B = 2x - 2\] and \[A - B = 4x - 8\], how do you find A and B?

Answer 1

Hint: Solve for \[A\] and \[B\] with the use of either elimination method or substitution method by assuming \[A\] and \[B\] as variables and equations as linear equations in two variables of \[A\] and \[B\].

Complete step by step solution:
Write the given equations and name them as shown below.
\[A + B = 2x - 2\] …… (1)
\[A - B = 4x - 8\] …… (2)

Assume linear expressions \[A\] and \[B\] as variables in the equation (1) and (2).

It is observed that equation (1) and equation (2) are linear equations in two variables \[A\] and \[B\].

Solve the linear equations pair by the use of elimination method, in which we eliminate one of the variables and solve for the other.

Add equation (1) and equation (2) to eliminate the variable \[B\] as shown below.
\[\begin{array}{l}A + B = 2x - 2\\\underline {A - B = 4x - 8} \\\,\,\,\,\,2A\,\,\, = 6x - 10\end{array}\]

Further solve the obtained equation \[2A = 6x - 10\] as shown below.

Divide by \[2\] from both sides of the equation and obtain the linear expression for \[A\] as,
\[\dfrac{{2A}}{2} = \dfrac{{6x - 10}}{2}\]
\[ \Rightarrow A = \dfrac{{6x}}{2} - \dfrac{{10}}{2}\]
\[ \Rightarrow A = 3x - 5\]

So, the linear expression for \[A\] is \[3x - 5\].

Now, substitute \[3x - 5\] for \[A\] in the equation (1) and obtain the linear expression for \[B\] as shown below.
\[A + B = 2x - 2\]
\[ \Rightarrow \left( {3x - 5} \right) + B = 2x - 2\]
\[ \Rightarrow 3x - 5 + B = 2x - 2\]

Simplify the equation by subtracting \[3x\] from both sides of the above equation as shown below.
\[ \Rightarrow 3x - 5 + B - 3x = 2x - 2 - 3x\]
\[ \Rightarrow - 5 + B = - 2 - x\]

Further, simplify the equation by addition of \[5\] from both sides of the above equation as shown below.
\[ \Rightarrow - 5 + B + 5 = - 2 - x + 5\]
\[ \Rightarrow B = 3 - x\]

So, the linear expression for \[B\] is \[3 - x\].
Thus, the linear expressions of \[A\] is \[3x - 5\] and the linear expression of \[B\] is \[3 - x\].

Note: The other way to obtain linear expressions for \[A\] and \[B\] is by the use of substitution method.

From equation (1), isolate variable \[A\] in terms of \[B\] as shown below.
\[A + B = 2x - 2\]
\[ \Rightarrow A = 2x - 2 - B\]

Now substitute this expression in equation (2) and obtain the expression \[B\] in terms of \[x\] as follow:
\[A - B = 4x - 8\]
\[ \Rightarrow \left( {2x - 2 - B} \right) - B = 4x - 8\]
\[ \Rightarrow 2x - 2 - B - B = 4x - 8\]
\[ \Rightarrow 2x - 2 - 2B = 4x - 8\]

Simplify the equation as,
\[ \Rightarrow 2x - 2 - 4x + 8 = 2B\]
\[ \Rightarrow - 2x + 6 = 2B\]
\[ \Rightarrow 2\left( { - x + 3} \right) = 2B\]
\[ \Rightarrow - x + 3 = B\]

So, the linear expression for \[B\] is \[3 - x\].

Substitute the expression of \[B\] in equation (1) and obtain the expression of \[A\] as shown below.
\[A + \left( {3 - x} \right) = 2x - 2\]
\[ \Rightarrow A + 3 - x = 2x - 2\]
\[ \Rightarrow A = 2x - 2 - 3 + x\]
\[ \Rightarrow A = 3x - 5\]

So, the linear expression for \[A\] is \[3x - 5\].

Therefore, the linear expressions of \[A\] is \[3x - 5\] and the linear expression of \[B\] is \[3 - x\].

Always solve these types of questions either by elimination method or substitution method. Choose the appropriate method depending on the linear equations pair.