Solve the following system. Enter the values for A and B as reduced fr

Tammy Fisher 2021-11-21 Answered
Solve the following system. Enter the values for A and B as reduced fractions or integers.
\(\displaystyle-{6}{A}+{8}{B}={48}\)
\(\displaystyle-{6}{A}+{2}{B}={42}\)
- One or more solutions:
- No solution
- Infinite number of solutions

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Expert Answer

Nancy Johnson
Answered 2021-11-22 Author has 670 answers
Step 1
Consider the system of equation
\(\displaystyle-{6}{A}+{8}{B}={48}\).....(1)
\(\displaystyle-{6}{A}+{2}{B}={42}\).....(2)
Subtract equation (2) from equation (1)
\(\displaystyle-{6}{A}+{8}{B}-{\left(-{6}{A}+{2}{B}\right)}={48}-{42}\)
\(\displaystyle-{6}{A}+{8}{B}+{6}{A}-{2}{B}={48}-{42}\)
\(\displaystyle{6}{B}={6}\) (Combine the like term)
Divide both sides by 6 and further simplify it
\(\displaystyle{\frac{{{6}{B}}}{{{6}}}}={\frac{{{6}}}{{{6}}}}\)
\(\displaystyle{B}={1}\)
Step 2
Substitute \(\displaystyle{B}={1}\) into \(\displaystyle-{6}{A}+{8}{B}={48}\)
\(\displaystyle-{6}{A}+{8}\times{1}={48}\)
\(\displaystyle-{6}{A}+{8}={48}\)
Subtract 8 from sides and further simplify it
\(\displaystyle-{6}{A}+{8}-{8}={48}-{8}\)
\(\displaystyle-{6}{A}={40}\)
Divide both sides by -6 and further simplify it
\(\displaystyle{\frac{{-{6}{A}}}{{-{6}}}}={\frac{{{40}}}{{-{6}}}}\)
\(\displaystyle{A}=-{\frac{{{20}}}{{{3}}}}\)
Therefore, \(\displaystyle{A}=-{\frac{{{20}}}{{{3}}}}\) and \(\displaystyle{B}={1}\)
Hence, the given system of equation has only one solution.
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