Question

graph the following system of equations: \\(\

$$\begin{cases} y = 5x - 1 \\\\ -10x + 5y = 25 \\end{cases}$$

\\) solution: \\(\underline{\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad}\\).

Explanation:

Step1: Simplify the second equation

We have the second equation \(-10x + 5y = 25\). Divide each term by \(5\) to simplify: \(\frac{-10x}{5}+\frac{5y}{5}=\frac{25}{5}\), which gives \(-2x + y = 5\). Then, solve for \(y\): \(y = 2x + 5\).

Step2: Analyze the slopes of the two lines

The first equation is \(y = 5x - 1\), so its slope \(m_1 = 5\). The second equation (after simplification) is \(y = 2x + 5\), so its slope \(m_2 = 2\). Since \(m_1
eq m_2\), the two lines are not parallel and will intersect at one point.

Step3: Solve the system using substitution

Substitute \(y = 5x - 1\) into the second original equation \(-10x + 5y = 25\). So we get \(-10x + 5(5x - 1)=25\). Expand the left - hand side: \(-10x+25x - 5 = 25\). Combine like terms: \(15x-5 = 25\). Add \(5\) to both sides: \(15x=25 + 5=30\). Divide both sides by \(15\): \(x = 2\).

Step4: Find the value of \(y\)

Substitute \(x = 2\) into the first equation \(y = 5x - 1\). Then \(y=5\times2-1=10 - 1 = 9\).

Answer:

The solution of the system of equations is \(x = 2,y = 9\) (or the ordered pair \((2,9)\))