Chapter A.6, Problem 127AYU

a.

To determine

Which of the following pairs of equations are equivalent?

Answer to Problem 127AYU

  $x^2=9$ and $x=3$ are not equivalent.

Explanation of Solution

Given information:

Which of the following pairs of equations are equivalent? Explain.

  $x^2=9;x=3$

Calculation: Determine if the pair of equations $x^2=9$ and $x=3$ are equivalent.

To see if the two equations are equal, solve the first equation for $x$

  $x^2=9$

  $\sqrt{x^2}=\sqrt{9}$

  $x=\pm \sqrt{3^2}$

  $x=\pm 3$

Hence, the solution to the equation $x^2=9$ are $x=-3$ and $x=3$ , the equations $x^2=9$ and $x=3$

are not equivalent.

b.

To determine

Which of the following pairs of equations are equivalent?

Answer to Problem 127AYU

  $x=\sqrt{9}$ and $x=3$ are equivalent.

Explanation of Solution

Given information:

Which of the following pairs of equations are equivalent? Explain.

  $x=\sqrt{9};x=3$

Calculation: Determine if the pair of equations $x=\sqrt{9}$ and $x=3$ are equivalent.

To see if the two equations are equal, solve the first equation for $x$

  $x=\sqrt{9}$

  $x=\sqrt{3^2}$

  $x=3$

Hence, the equation $x=\sqrt{9}$ simplifies to $x=3$ , the equations $x=\sqrt{9}$ and $x=3$ in (b) are equivalent.

c.

To determine

Which of the following pairs of equations are equivalent?

Answer to Problem 127AYU

  $\left(x-1\right)\left(x-2\right)={\left(x-1\right)}^2$ and $x-2=x-1$ are not equivalent.

Explanation of Solution

Given information:

Which of the following pairs of equations are equivalent? Explain.

  $\left(x-1\right)\left(x-2\right)={\left(x-1\right)}^2;x-2=x-1$

Calculation: Determine if the pair of equations $\left(x-1\right)\left(x-2\right)={\left(x-1\right)}^2$ and $x-2=x-1$ are equivalent.

To see it the two equations are equal, solve the first for $x$

  $\left(x-1\right)\left(x-2\right)={\left(x-1\right)}^2$

  $x^2-3x+2=x^2-2x+1$

  $-3x+2=-2x+1$

  $1=x$

So the equation $\left(x-1\right)\left(x-2\right)={\left(x-1\right)}^2$ simplifies to $x=1$ .

Now simplify $x-2=x-1$ to see if we get an equivalent equation

  $x-2=x-1$

  $x-2-x=x-1-x$

  $-2\ne -1$

In the second equation we do not even get a true statement.

Hence, the equations $\left(x-1\right)\left(x-2\right)={\left(x-1\right)}^2$ and $x-2=x-1$ are not equivalent.