Questions: Problem 7. (1 point) For what value(s) of (h) and (k) does the linear system have infinitely many solutions? [ 7 x1+2 x2 =-1 h x1+k x2 =2 ] (h) and (k) Note: (h) and (k) must be integers or fractions reduced to lowest terms.

$Problem 7. (1 point) For what value(s) of (h) and (k) does the linear system have infinitely many solutions? [ 7 x1+2 x2 =-1 h x1+k x2 =2 ] (h) and (k) Note: (h) and (k) must be integers or fractions reduced to lowest terms.$

Transcript text: Problem 7. (1 point) For what value(s) of $h$ and $k$ does the linear system have infinitely many solutions? \[ \begin{aligned} 7 x_{1}+2 x_{2} & =-1 \\ h x_{1}+k x_{2} & =2 \end{aligned} \] h $\square$ and $k$ $\square$ $\square$ Note: $h$ and $k$ must be integers or fractions reduced to lowest terms.

Solution

Solution Steps

To have infinitely many solutions, the two equations must be scalar multiples of each other. This means that the ratios of the coefficients of $x_1$, $x_2$, and the constants on the right side must be equal. We will set up these ratios and solve for $h$ and $k$.

Step 1: Set Up the Condition for Infinitely Many Solutions

For the system of linear equations to have infinitely many solutions, the second equation must be a scalar multiple of the first. This means the ratios of the coefficients of $x_1$, $x_2$, and the constants must be equal. Therefore, we set up the following equations: \[ \frac{h}{7} = \frac{k}{2} = \frac{2}{-1} \]

Step 2: Solve the Ratios

We solve the equations: