{:[y=0.25 x+12],[y=K(x+3)]:} In the system of equations, K is a constant. For which value of K does the system have no solution? Choose 1 answer: (A) 0 (B) 0.25 (c) 0.75 (D) 12

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{:[y=0.25 x+12],[y=K(x+3)]:} In the system of equations,  K is a constant. For which value of  K does the system have no solution? Choose 1 answer: (A) 0 (B) 0.25 (c) 0.75 (D) 12

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Determining Slopes: To determine the value of K for which the system of equations has no solution, we need to look at the slopes of the two lines represented by the equations. If the slopes are the same and the y-intercepts are different, the lines are parallel and will never intersect, meaning there is no solution to the system. The first equation is y = 0.25x + 12, which is in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. The slope of the first line is 0.25.Comparing Equations: The second equation is y = K(x + 3). To compare it to the first equation, we need to distribute K to both terms inside the parentheses to get it into the form y = mx + b. Doing this, we get y = Kx + 3K.Finding the Value of K: Now we can see that the slope of the second line is K. For the system to have no solution, the slope of the second line must be the same as the slope of the first line, which is 0.25. Therefore, K must be 0.25.Checking Y-Intercepts: However, we also need to ensure that the y-intercepts are different. The y-intercept of the first equation is 12. For the second equation, the y-intercept is 3K. If K is 0.25, then 3K would be 3 * 0.25 = 0.75, which is different from 12. This confirms that the lines are parallel and will not intersect, meaning the system has no solution.

$\begin{array}{l} y=0.25 x+12 \\ y=K(x+3) \end{array}$ $\newline$ In the system of equations, $K$ is a constant. For which value of $K$ does the system have no solution? $\newline$ Choose $1$ answer: $\newline$ (A) $0$ $\newline$ (B) $0$ . $25$ $\newline$ (C) $0$ . $75$ $\newline$ (D) $12$

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