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ay=2x+1
y=2x+2
Consider the system of equations, where a is a constant. For what value of a are there no (x,y) solutions?
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$ay=2x+1$ $\newline$ $y=2x+2$ $\newline$ Consider the system of equations, where $a$ is a constant. For what value of $a$ are there no $(x,y)$ solutions? $\newline$ $\square$

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Find the constant of variation

Full solution

Q.

ay=2x+1

\newline

y=2x+2

\newline

Consider the system of equations, where

a

is a constant. For what value of

a

are there no

(x,y)

solutions?

\newline

\square

Identify System: The system of equations is given by $ay = 2x + 1$ and $y = 2x + 2$ . To find the value of $a$ for which there are no solutions, we need to look for a condition that would make the system inconsistent.
Substitute $y$ : Since the second equation is $y = 2x + 2$ , we can substitute this expression for $y$ into the first equation to get $a(2x + 2) = 2x + 1$ .
Expand Equation: Expanding the left side of the equation, we get $2ax + 2a = 2x + 1$ .
Set Coefficients: For the system to have no solutions, the lines represented by the equations must be parallel. This means the coefficients of $x$ must be the same, and the constant terms must be different. Therefore, we set $2a$ equal to $2$ and $2a$ not equal to $1$ .
Solve for $a$ : Solving $2a = 2$ gives us $a = 1$ . However, for no solutions, we also need $2a \neq 1$ . Since $2\times1 = 2$ , which is not equal to $1$ , we have a contradiction. Therefore, the value of $a$ that makes the system inconsistent is $a = 1$ .
Conclude Solution: We conclude that when $a = 1$ , the system of equations has no solutions because the lines are parallel and have different $y$ -intercepts.

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Question

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\newline

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