Question

8.4.14
date:
period:

1. here is a system of equations: $\begin{cases} x = 14 \\ 2x - 5y = 13 end{cases}$

in the solution $(x, y)$, what is the value of $y$?

1. here is an incomplete system of equations. create a second equation so that the system has no solution.

$\begin{cases} y = \frac{3}{4}x - 4 \\ \boldsymbol{?} end{cases}$

Explanation:

Step1: Substitute $x=14$ into second equation

Substitute $x=14$ into $2x-5y=13$:
$2(14)-5y=13$

Step2: Calculate the product

Compute $2\times14$:
$28-5y=13$

Step3: Isolate the $y$-term

Subtract 28 from both sides:
$-5y=13-28$
$-5y=-15$

Step4: Solve for $y$

Divide both sides by $-5$:
$y=\frac{-15}{-5}=3$

Step5: Define no-solution system rule

A system has no solution if equations are parallel (same slope, different y-intercept).

Step6: Create parallel equation

Given $y=\frac{3}{4}x-4$, use same slope $\frac{3}{4}$, different intercept:
$y=\frac{3}{4}x+1$ (any non -4 intercept works)

Answer:

1. $y=3$
2. $y=\frac{3}{4}x+1$ (example; any equation with slope $\frac{3}{4}$ and y-intercept not equal to $-4$ is valid)