Question

which system of equations is best solved by substitution?
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$$\begin{cases}x + y = 10 \\\\ x - y = 2\\end{cases}$$

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$$\begin{cases}3x + 2y = 8 \\\\ 2x - 3y = 1\\end{cases}$$

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$$\begin{cases}2x + 3y = 12 \\\\ 4x - 3y = 6\\end{cases}$$

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$$\begin{cases}y = 4x - 5 \\\\ 2x + y = 7\\end{cases}$$

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Explanation:

Substitution method is best when one variable is already expressed in terms of the other (or can be easily expressed). Let's analyze each system:

System 1: \( x + y = 10 \), \( x - y = 2 \)

We can solve for \( x \) or \( y \), but let's check others.

System 2: \( 3x + 2y = 8 \), \( 2x - 3y = 1 \)

Both equations are in standard form, elimination might be easier.

System 3: \( 2x + 3y = 12 \), \( 4x - 3y = 6 \)

Elimination (adding equations to eliminate \( y \)) is easier here.

System 4: \( y = 4x - 5 \), \( 2x + y = 7 \)

Here, \( y \) is already expressed as \( 4x - 5 \). We can substitute \( y = 4x - 5 \) into the second equation \( 2x + y = 7 \) directly. Let's verify:

Substitute \( y = 4x - 5 \) into \( 2x + y = 7 \):
\( 2x + (4x - 5) = 7 \)
\( 6x - 5 = 7 \)
\( 6x = 12 \)
\( x = 2 \)
Then \( y = 4(2) - 5 = 3 \)

This system is ideal for substitution because one equation gives \( y \) in terms of \( x \), making substitution straightforward.

Answer:

The system \( \boldsymbol{y = 4x - 5}\) and \( \boldsymbol{2x + y = 7}\) (the fourth system) is best solved by substitution.