{:[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],[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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Identify Inconsistent Condition: 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. This would occur if the two equations represent parallel lines, which means they have the same slope but different y-intercepts.Convert Second Equation: The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. The second equation is already in slope-intercept form with a slope of 2. To compare it with the first equation, we need to express the first equation in terms of y.Express First Equation in y: To express the first equation in terms of y, we divide both sides by a to get y = (2/a)x + 1/a.Set Slopes Equal: Now we have y = (2/a)x + 1/a and y = 2x + 2. For the lines to be parallel, the slopes must be equal, which means 2/a must be equal to 2. Setting these equal gives us 2/a = 2.Solve for a: Solving for a, we multiply both sides by a to get 2 = 2a. Then we divide both sides by 2 to get a = 1.Analyze Solutions: However, if a = 1, then the two equations are actually the same line (since they would both have the same slope and y-intercept), which means there would be infinitely many solutions. Therefore, for there to be no solutions, the slopes must be the same, but the y-intercepts must be different. Since the slopes are already the same when a = 1, we need to find a value of a that does not change the slope but makes the y-intercept different. This is not possible because any value of a other than 1 would change the slope. Therefore, there is no value of a that would result in no solutions for the given system of equations.

$\begin{array}{r} a y=2 x+1 \\ y=2 x+2 \end{array}$ $\newline$ 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+1y=2x+2 \begin{array}{r} a y=2 x+1 \\ y=2 x+2 \end{array} ay=2x+1y=2x+2​\newlineConsider the system of equations, where a a a is a constant. For what value of a a a are there no (x,y) (x, y) (x,y) solutions?

$\begin{array}{r} a y=2 x+1 \\ y=2 x+2 \end{array}$ $\newline$ Consider the system of equations, where $a$ is a constant. For what value of $a$ are there no $(x, y)$ solutions?