Question

Use substitution to solve the system. $$\begin{aligned}&4 x-y=5\\\&2 x+4 y=15\end{aligned}$$

Short answer

The solution is \( ( x, y) = (35/18, 5/9) \)

Step-by-step solution

Rearrange the first equation

The first step is to rearrange the first equation \(4x - y = 5\) to express y in terms of x. This can be done by adding y to both sides and subtracting 5 from both sides: \( y = 4x - 5 \)

Substitute \(y\) in the second equation

Substitute \(y = 4x - 5\) in the second equation \(2x + 4y = 15\), to get an equation only in terms of x: \(2x + 4(4x - 5) = 15\).

Solve for \(x\)

Simplify the equation to solve for \(x\): \(2x + 16x - 20 = 15\), then simplify further to get \(18x = 35\), which can be solved for \(x\) by dividing both sides by 18, to get \(x = 35 / 18\).

Substitute \(x\) back to get value for \(y\)

Substitute \(x = 35 / 18\) back into equation \(y = 4x - 5\) to solve for \(y\), which gives \(y = 4(35/18) - 5 = 5 / 9\).

Interpret the solution

The solution to the system of equations is the point \(( x, y) = (35/18, 5/9)\), which means that both of the given equations would be satisfied at this point.