Question

practice & problem solving
leveled practice in 7 and 8, complete the equations to find the number of solutions.

1. classify the equation 33x + 99 = 33x - 99 as having one solution, no solution, or infinitely many solutions.

33x + 99 = 33x - 99
33x - + 99 = 33x - - 99
99 - 99
since 99 is equal to - 99, the equation has solution(s).

1. solve 4(4x + 3)=19x + 9 - 3x + 3. does the equation have one solution, no solution, or infinitely many solutions?

4(4x + 3)=19x + 9 - 3x + 3
4· + 4· = 19x + 9 - 3x + 3
16x + 12 = +
16x - + 12 = 16x - + 12
12 12
since 12 is equal to 12, the equation has solution(s).

1. generalize what does it mean if an equation is equivalent to 0 = 0? explain.
2. solve 4x + x + 4 = 8x - 3x + 4. does the equation have one solution, no solution, or infinitely many solutions? if one solution, write the solution. explain.
3. reasoning two rival dry cleaners both advertise their prices. let x equal the number of items dry cleaned. store as prices are represented by the expression 15x - 2. store bs prices are represented by the expression 3(5x + 7). when do the two stores charge the same rate? explain.

Explanation:

Step1: Solve equation 7

Given $33x + 99=33x - 99$. Subtract $33x$ from both sides:
$33x-33x + 99=33x-33x - 99$, which simplifies to $99=-99$. Since $99$ is not equal to $-99$, the equation has no solution.

Step2: Solve equation 8

Given $4(4x + 3)=19x + 9-3x + 3$. First, expand the left - hand side using the distributive property: $4\times4x+4\times3=16x + 12$.
Simplify the right - hand side: $19x-3x + 9 + 3=16x+12$.
So we have $16x + 12=16x + 12$. Subtract $16x$ from both sides: $16x-16x + 12=16x-16x + 12$, which gives $12 = 12$. Since $12$ is equal to $12$, the equation has infinitely many solutions.

Step3: Solve equation 9

If an equation is equivalent to $0 = 0$, it means that the equation is an identity. An identity is true for all values of the variable. So the equation has infinitely many solutions.

Step4: Solve equation 10

Given $4x+x + 4=8x-3x + 4$. Combine like terms on both sides.
Left - hand side: $4x+x=5x$, so it is $5x + 4$.
Right - hand side: $8x-3x=5x$, so it is $5x + 4$.
We get $5x + 4=5x + 4$. Subtract $5x$ from both sides: $5x-5x + 4=5x-5x + 4$, which gives $4 = 4$. Since $4$ is equal to $4$, the equation has infinitely many solutions.

Step5: Solve equation 11

Set the two price expressions equal to each other: $15x-2=3(5x + 7)$.
Expand the right - hand side: $15x-2=15x+21$.
Subtract $15x$ from both sides: $15x-15x-2=15x-15x + 21$, which simplifies to $-2 = 21$. Since $-2$ is not equal to $21$, the two stores never charge the same rate.

Answer:

1. No solution
2. Infinitely many solutions
3. The equation is an identity and has infinitely many solutions
4. Infinitely many solutions
5. The two stores never charge the same rate.