Question

3 Let $\beta$ be any number and consider the system of equations: $x - \beta y = 1$ $\beta x - y = -1$ For which values of $\beta$ is the system (i) inconsistent (ii) consistent with a unique solution (iii) consistent with infinitely many solutions? [Hint: row reduce the matrix as normal, but be careful if you come to a point where you might be dividing by zero. For example, if you wanted to divide a row by $k$, you need to divide into two cases: (i) $k \neq 0$ when you can divide through by $k$ and continue, and (ii) $k = 0$ when you can replace all $ks$ by zero and continue.]

          3 Let $\beta$ be any number and consider the system of equations:
$x - \beta y = 1$
$\beta x - y = -1$

For which values of $\beta$ is the system (i) inconsistent (ii) consistent with a unique
solution (iii) consistent with infinitely many solutions?

[Hint: row reduce the matrix as normal, but be careful if you come to a point
where you might be dividing by zero. For example, if you wanted to divide a
row by $k$, you need to divide into two cases: (i) $k \neq 0$ when you can divide
through by $k$ and continue, and (ii) $k = 0$ when you can replace all $ks$ by zero
and continue.]

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3 Let β be any number and consider the system of equations:
x - β y = 1
β x - y = -1

For which values of β is the system (i) inconsistent (ii) consistent with a unique
solution (iii) consistent with infinitely many solutions?

[Hint: row reduce the matrix as normal, but be careful if you come to a point
where you might be dividing by zero. For example, if you wanted to divide a
row by k, you need to divide into two cases: (i) k ≠ 0 when you can divide
through by k and continue, and (ii) k = 0 when you can replace all ks by zero
and continue.]

Added by Kathleen W.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Aparna Shakti

Step 1

Let's start by multiplying the second equation by Î²: Î²(Î²x - y) = -1 Î²(Î²x) - Î²y = -Î² Now, let's subtract the first equation from this new equation: (Î²(Î²x) - Î²y) - (x - Î²y) = (-Î²) - 1 (Î²^2x - Î²y) - x + Î²y = -Î² - 1 (Î²^2 - 1)x = -Î² - 1 Now, we Show more…

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Let Î² be any number and consider the system of equations: x - Î²y = 1 Î²x - y = -1 For which values of Î² is the system (i) inconsistent, (ii) consistent with a unique solution, and (iii) consistent with infinitely many solutions? [Hint: row reduce the matrix as normal, but be careful if you come to a point where you might be dividing by zero. For example, if you wanted to divide a row by k, you need to divide into two cases: (i) k â‰ 0 when you can divide through by k and continue, and (ii) k = 0 when you can replace all ks by zero and continue.] Let Î± be any number and consider the system of equations: x + 3x = 3y 3y - Î±y = 1 For which values of Î± is the system (i) inconsistent, (ii) consistent with a unique solution, and (iii) consistent with infinitely many solutions? Hint: row reduce the matrix as normal, but be careful if you come to a point where you might be dividing by zero. For example, if you wanted to divide a row by k, you need to divide into two cases: (i) k â‰ 0 when you can divide through by k and continue, and (ii) k = 0 when you can replace all ks by zero and continue.]

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