Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

If $a$ and $b$ are non-zero integers such that $a^2+b^2-4a-2b=0$ and $a^2-b^2=0$ , which of the following could be the value of $(a-b)$ ? $\newline$ (A) $-3$ $\newline$ (B) $2$ $\newline$ (C) $3$ $\newline$ (D) $6$

View step-by-step help

View step-by-step help

Transformations of linear functions

Full solution

Q. If

a

and

b

are non-zero integers such that

a^2+b^2-4a-2b=0

and

a^2-b^2=0

, which of the following could be the value of

(a-b)

?

\newline

(A)

-3

\newline

(B)

2

\newline

(C)

3

\newline

(D)

6

Identify Equations and Relationship: Identify the given equations and the relationship between $a$ and $b$ . We have two equations: $1$ . $a^2 + b^2 − 4a − 2b = 0$ $2$ . $a^2 − b^2 = 0$ The second equation can be factored using the difference of squares. $a^2 − b^2 = (a + b)(a − b) = 0$ Since $a$ and $b$ are non-zero integers, $a + b$ cannot be zero, so $a − b$ must be zero.
Determine Value of a: Determine the value of a in terms of b using the second equation. $\newline$ From $a^2 - b^2 = 0$ , we have: $\newline$ $a - b = 0$ $\newline$ $a = b$
Substitute Value of $a$ : Substitute the value of $a$ into the first equation. $\newline$ Since $a = b$ , we can replace $a$ with $b$ in the first equation: $\newline$ $b^2 + b^2 − 4b − 2b = 0$ $\newline$ Combine like terms: $\newline$ $2b^2 − 6b = 0$
Factor Out $b$ : Factor out $b$ from the equation. $b(2b − 6) = 0$ Since $b$ is a non-zero integer, we can't have $b = 0$ , so we must have: $2b − 6 = 0$
Solve for b: Solve for b. $\newline$ Add $6$ to both sides of the equation: $\newline$ $2b = 6$ $\newline$ Divide both sides by $2$ : $\newline$ $b = 3$
Find Value of a: Since $a = b$ , find the value of $a$ . $\newline$ $a = b = 3$
Calculate $(a - b)$ : Calculate the value of $(a − b)$ . $\newline$ Since $a = b = 3$ , we have: $\newline$ $(a − b) = 3 − 3 = 0$ $\newline$ However, this contradicts the options given, as none of them are zero. We need to recheck our steps for any errors.

More problems from Transformations of linear functions

Question

This equation shows how the time required to screen-print a batch of shirts is related to the number of shirts in the batch.

\newline

t = s + 13

\newline

s

represents the number of shirts in the batch, and the variable

t

represents the time required to screen-print the shirts. How long does it take to screen-print a batch of

4

\newline

_________minutes

Posted 3 months ago

Question

Krysta wants to take piano lessons. She asked a teacher how many new pieces she could expect to learn each month. The teacher gave an estimate of

3

. Write an equation that shows how the total number of pieces Krysta would learn,

y

, depends on the number of months of piano lessons,

x

\newline

y

Posted 3 months ago

Question

Tom bought a boat for

\$3,000

and incurs a daily storage fee of

\$15

. Write a linear function to represent the total amount Tom will owe,

p(x)

x

days to pick it up.

\newline

Posted 4 months ago

Question

180

-page book cover to cover in a single session, at a constant rate. It took him

6

\newline

\newline

y

represent the number of pages left to read after

\newline

t

\newline

Which of the following information about the graph of the relationship is given?

\newline

1

\newline

\newline

x

\newline

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

\newline

x

\newline

y

\newline

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 4 months ago

Question

A charity organization had to sell

18

tickets to their fundraiser just to cover necessary production costs. They sold each ticket for

\$45

y

represent the net profit (in dollars) when they have sold

x

tickets. Which of the following information about the graph of the relationship is given?

\newline

1

\newline

x

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

x

y

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 2 months ago

Question

Hiro painted his room. After

3

hours of painting at a rate of

8

square meters per hour, he had

28

square meters left to paint.

\newline

\newline

y

represent the area (in square meters) left to paint after

\newline

x

\newline

Which of the following information about the graph of the relationship is given?

\newline

1

\newline

\newline

x

\newline

\newline

y

\newline

(C) Slope and a point that is not an intercept

\newline

\newline

x

\newline

y

\newline

\newline

y

-intercept and a point that is not an intercept

\newline

(F) Two points that are not intercepts

Posted 2 months ago

Question

Naoya read a book cover to cover in a single session, at a rate of

55

pages per hour. After

4

350

pages left to read.

\newline

y

represent the number of pages left to read after

x

\newline

Complete the equation for the relationship between the number of pages left and number of hours.

\newline

y=

Posted 1 month ago

Question

A teacher purchased

300

colored pencils for an upcoming project. The pencils she ordered come in packs of

20

30

14

packs in all. Write a system of equations to find the number of

20

(x)

30

(y)

the art teacher ordered.

Posted 2 months ago

Question

C=0.25(L-1200)

\newline

C

, in dollars, to the holder of a liability insurance account given a total liability of

L

dollars is given by the equation as long as

C

is positive. By how many dollars does the liability increase when the cost to the holder increases by

1

Posted 2 months ago

Question

P=\frac{2}{5} m+1.5

\newline

P

, in dollars, of a train ticket used to travel a distance of

m

miles is given by the equation. By how many miles does the traveling distance increase if the ticket price increases by

1

Posted 4 months ago