Warm Up Solutions

Solve the systems of linear equations.

$a = \frac{1}{2} b 7 a - 5 b = - 105$

$\begin{array}{rcl} 7 (\frac{1}{2} b) - 5 b & = & - 105 \\ (2) (\frac{7}{2} b - 5 b & = & \frac{}{} - 105) \\ 7 b - 10 b & = & - 210 \\ - 3 b & = & - 210 \\ b & = & 70 \\ a & = & \frac{1}{2} (70) \end{array}$

$a = 35$

Note

This system is solved by clearing fractions. Equality is maintained when EVERY term on BOTH SIDES is multiplied by the same factor.

Q: How do you write the answer to a system of equations with one solution?
A: As an ordered pair, with the variables in alphabetical order (i.e., (a, b))

$5 x - 3 y = 9 3 x + 3 y = 5$

$\begin{array}{rcl} add equations vertically \\ 8 x = 14 x = \frac{7}{4} 3 (\frac{7}{4}) + 3 y = 5 (4) (\frac{21}{4} + 3 y = 5) 21 + 12 y = 20 12 y = - 1 y = - \frac{1}{12} \end{array}$

$(\frac{7}{4}, - \frac{1}{12})$

Note

Q: How can you check a solution to a system?
A: Substitute the ordered pair into BOTH equations.