Question

1. ellens math homework involves finding out how many solutions some equations have. drag each equation on the left to the number of solutions it has. choices: 5(x + 1)=25; 12 + 3x+5=x + 19+2x - 2; 4x=3(x + 4)-8; 2x + 10+6x=8x - 7. answers: no solution; exactly one solution; infinitely many solutions. 6. what is the value of x in the equation 0.2 + 0.4x-0.3(x - 10)=0.3x - 4? 7. triangle abc is translated 5 units up and 3 units to the left to obtain the image abc. which of the following statements about this translation are true? select all that apply. a. the coordinates of a would be (0,9). b. the length of bc would be 12 units. c. the coordinates of c would be (5,-10). d. the measure of angle b would be less than the measure of angle b. e. the coordinates of b would be (-5,-2).

Explanation:

5.

Step1: Simplify 5(x + 1)=25

Expand the left - hand side: $5x+5 = 25$. Then subtract 5 from both sides: $5x=20$, and divide by 5 to get $x = 4$. So it has exactly one solution.

Step2: Simplify 12 + 3x+5=x + 19+2x - 2

Combine like - terms: $3x + 17=3x+17$. This is an identity, so it has infinitely many solutions.

Step3: Simplify 4x=3(x + 4)-8

Expand the right - hand side: $4x=3x + 12-8$, then $4x=3x + 4$. Subtract $3x$ from both sides to get $x = 4$. So it has exactly one solution.

Step4: Simplify 2x+10 + 6x=8x - 7

Combine like - terms: $8x+10=8x - 7$. Subtract $8x$ from both sides: $10=-7$, which is a contradiction. So it has no solution.

Step1: Expand the equation 0.2+0.4x-0.3(x - 10)=0.3x - 4

$0.2+0.4x-0.3x + 3=0.3x - 4$.

Step2: Combine like - terms

$(0.4x-0.3x)+(0.2 + 3)=0.3x - 4$, so $0.1x+3.2=0.3x - 4$.

Step3: Move the x terms to one side and constants to the other side

Subtract $0.1x$ from both sides: $3.2=0.3x-0.1x - 4$, then $3.2 = 0.2x-4$. Add 4 to both sides: $3.2 + 4=0.2x$, so $7.2=0.2x$.

Step4: Solve for x

Divide both sides by 0.2: $x=\frac{7.2}{0.2}=36$.

We need to know the original coordinates of the points of triangle ABC to check each option. But if we assume the general translation rule:

Step1: Analyze option A

Without knowing the original coordinates of A, we can't say for sure that the coordinates of $A'$ are $(0,9)$.

Step2: Analyze option B

Translation is a rigid motion, which preserves the lengths of line - segments. But without knowing the original length of BC, we can't say that the length of $B'C'$ is 12 units.

Step3: Analyze option C

Without knowing the original coordinates of C, we can't say for sure that the coordinates of $C'$ are $(5,-10)$.

Step4: Analyze option D

Translation is a rigid motion that preserves angle measures. So the measure of angle $B'$ is equal to the measure of angle B, not less.

Step5: Analyze option E

Without knowing the original coordinates of B, we can't say for sure that the coordinates of $B'$ are $(-5,-2)$.

However, if we assume we have enough information about the original coordinates:
Let the original coordinates of point B be $(x,y)$. After translation, the new coordinates of $B'$ are $(x-3,y + 5)$.

If we assume the original coordinates of B are $( - 2,-7)$:
The new coordinates of $B'$ are $(-2-3,-7 + 5)=(-5,-2)$

Answer:

No solution: $2x + 10+6x=8x - 7$
Exactly one solution: $5(x + 1)=25$, $4x=3(x + 4)-8$
Infinitely many solutions: $12 + 3x+5=x + 19+2x - 2$

6.