Question

Use linear combinations to solve the system of linear equations. $$\begin{array}{l} 3 a+9 b=8 b-a \\ 5 a-10 b=4 a-9 b+5 \end{array}$$

Short answer

The solutions to the system of equations are \(a = 1\) and \(b = -4\).

Step-by-step solution

Simplify the equations

First, simplify each of the given equations: For Equation 1: \(3a + 9b = 8b - a\) This simplifies to: \(4a + b = 0\).For Equation 2: \(5a - 10b = 4a - 9b + 5\) This simplifies to: \(a - b = 5\).

Solve for a

From the simplified form of Equation 2: \(a - b = 5\), we can solve for a by rearranging the equation \(a = b + 5\)

Substitute a in Equation 1

Substitute \(a = b + 5\) into the simplified form of Equation 1: \(4a + b = 0\) to get: \(4(b + 5) + b = 0\). This simplifies to \(5b = -20\)

Solve for b

Solving \(5b = -20\), we get \(b = -4\)

Substitute b into Equation 2

Finally, substitute \(b = -4\) into Equation 2 to solve for a: \(a = -4 + 5 = 1\)