Question

solve this system of equations by using substitution.

\\\

$$\begin{cases} x = 2y \\\\ 4x + 2y = 30 \\end{cases}$$

\\

\\((?, \quad)\\)

Explanation:

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<reasoning>

Identify the given system of equations

We are given the following system of linear equations:
\[

$$\begin{cases} x = 2y & \text{(Equation 1)} \\ 4x + 2y = 30 & \text{(Equation 2)} \end{cases}$$

\]
Our goal is to find the solution as an ordered pair \((x, y)\).

Substitute Equation 1 into Equation 2

Using the Linear Equations knowledge point
\[

$$\begin{aligned} 4(2y) + 2y &= 30 \\ 8y + 2y &= 30 \end{aligned}$$

\]

Solve for y

Using the Linear Equations knowledge point
\[

$$\begin{aligned} 10y &= 30 \\ y &= 3 \end{aligned}$$

\]

Solve for x

Using the Linear Equations knowledge point
\[

$$\begin{aligned} x &= 2(3) \\ x &= 6 \end{aligned}$$

\]

Verify the solution

We substitute \(x = 6\) and \(y = 3\) back into Equation 2 to verify:
\[
4(6) + 2(3) = 24 + 6 = 30
\]
The solution is correct. The ordered pair is \((6, 3)\).
</reasoning>

<answer>
Solve this system of equations by using substitution.
\[

$$\begin{cases} x = 2y \\ 4x + 2y = 30 \end{cases}$$

\]
(<blank>6</blank>, <blank>3</blank>)
</answer>

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</post_analysis>

Answer:

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<reasoning>

Identify the given system of equations

We are given the following system of linear equations:
\[

$$\begin{cases} x = 2y & \text{(Equation 1)} \\ 4x + 2y = 30 & \text{(Equation 2)} \end{cases}$$

\]
Our goal is to find the solution as an ordered pair \((x, y)\).

Substitute Equation 1 into Equation 2

Using the Linear Equations knowledge point
\[

$$\begin{aligned} 4(2y) + 2y &= 30 \\ 8y + 2y &= 30 \end{aligned}$$

\]

Solve for y

Using the Linear Equations knowledge point
\[

$$\begin{aligned} 10y &= 30 \\ y &= 3 \end{aligned}$$

\]

Solve for x

Using the Linear Equations knowledge point
\[

$$\begin{aligned} x &= 2(3) \\ x &= 6 \end{aligned}$$

\]

Verify the solution

We substitute \(x = 6\) and \(y = 3\) back into Equation 2 to verify:
\[
4(6) + 2(3) = 24 + 6 = 30
\]
The solution is correct. The ordered pair is \((6, 3)\).
</reasoning>

<answer>
Solve this system of equations by using substitution.
\[

$$\begin{cases} x = 2y \\ 4x + 2y = 30 \end{cases}$$

\]
(<blank>6</blank>, <blank>3</blank>)
</answer>

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