$-6.4 x=4 y+2.1$ $\newline$ $k y+3.2 x=5.8$ $\newline$ For what value of $k$ does the system of linear equations in the variables $x$ and $y$ have no solutions?

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Algebra 2

Complementary angle identities

Full solution

-6.4 x=4 y+2.1

\newline

k y+3.2 x=5.8

\newline

For what value of

k

does the system of linear equations in the variables

x

and

y

have no solutions?

Identify Equations Form: Identify the form of the equations and the condition for no solutions. $\newline$ The given system of equations is in the form of a linear system. For a system of two linear equations to have no solutions, the lines represented by the equations must be parallel. This means that the ratios of the coefficients of $x$ and $y$ must be equal, while the constant terms are not proportional.
First Equation Slope: Write the first equation in slope-intercept form $y = mx + b$ . $\newline$ To find the slope of the first equation, we solve for $y$ : $\newline$ $-6.4x = 4y + 2.1$ $\newline$ $4y = -6.4x - 2.1$ $\newline$ $y = (-6.4/4)x - (2.1/4)$ $\newline$ $y = -1.6x - 0.525$ $\newline$ The slope of the first line is $-1.6$ .
Second Equation Slope: Write the second equation in slope-intercept form $y = mx + b$ . $\newline$ To find the slope of the second equation, we solve for $y$ : $\newline$ $ky + 3.2x = 5.8$ $\newline$ $ky = -3.2x + 5.8$ $\newline$ $y = \left(-\frac{3.2}{k}\right)x + \frac{5.8}{k}$ $\newline$ The slope of the second line is $-\frac{3.2}{k}$ .
Set Equal Slopes: Set the slopes of the two lines equal to each other to find the value of $k$ that makes the lines parallel. $\newline$ Since the slopes must be equal for the lines to be parallel, we have: $\newline$ $-1.6 = -3.2/k$ $\newline$ Now, solve for $k$ : $\newline$ $k = -3.2 / -1.6$ $\newline$ $k = 2$
Check Constant Terms: Check if the constant terms are not proportional. $\newline$ For the lines to have no solutions, the constant terms must not be proportional. We have the constant term of the first line as $-0.525$ and the second line as $\frac{5.8}{k}$ . Substituting $k = 2$ , we get: $\newline$ $\frac{5.8}{k} = \frac{5.8}{2} = 2.9$ $\newline$ Since $-0.525$ is not equal to $2.9$ , the constant terms are not proportional.