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{:[ax+by=1],[ax-by=1],[y=c]:}
In the system of three linear equations above,
a,b and
c are constants and
b!=0. If the system has exactly one solution, what is the value of
c in terms of
a and
b ?

$\begin{array}{l} a x+b y=1 \\ a x-b y=1 \\ y=c \end{array}$ $\newline$ In the system of three linear equations above, $a, b$ and $c$ are constants and $b \neq 0$ . If the system has exactly one solution, what is the value of $c$ in terms of $a$ and $b$ ?

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Write a linear equation from a slope and y-intercept

Full solution

Q.

\begin{array}{l} a x+b y=1 \\ a x-b y=1 \\ y=c \end{array}

\newline

In the system of three linear equations above,

a, b

and

c

are constants and

b \neq 0

. If the system has exactly one solution, what is the value of

c

in terms of

a

and

b

?

Identify Equations: Identify the system of equations. $\newline$ The system of equations is given by: $\newline$ \begin{cases}a x + b y = 1\a x - b y = 1\y = c\end{cases}
Conditions for Solution: Determine the conditions for a unique solution. $\newline$ For a system of linear equations to have exactly one solution, the equations must be independent and consistent. This means that no equation can be a multiple of another, and they must intersect at a single point.
Eliminate x: Subtract the second equation from the first to eliminate x. $\newline$ $(ax + by) - (ax - by) = 1 - 1$ $\newline$ This simplifies to: $\newline$ $2by = 0$
Solve for y: Solve for y. $\newline$ Since $b$ is not equal to $0$ ( $b \neq 0$ ), we can divide both sides by $2b$ to find $y$ . $\newline$ $y = \frac{0}{2b}$ $\newline$ $y = 0$
Compare with Third Equation: Compare the result with the third equation. $\newline$ The third equation states that $y = c$ . Since we found that $y = 0$ for the system to have a unique solution, we can equate $c$ to $0$ . $\newline$ $c = 0$

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