Question

1. solve for x: p = 2l + 2w
2. -7n = 28
3. a rectangles perimeter is 48 cm. its length is 5 cm more than its width. find the original dimensions.
4. -3(x - 1)+2(x + 5)=x + 7
5. 5÷1/5
6. select two that have infinitely many solutions:

a. 2(x - 3)=2x - 6
b. 3x + 2 = 3x - 2
c. -4(x + 5)=-4x - 20
d. x/2=(x + 4)/2+2

1. (48 - 18)/(15 - 3)

Explanation:

Step1: Solve for \(x\) in \(P = 2L+2W\) (There seems to be a mis - statement in the problem as there is no \(x\) in the equation. Assuming it was a wrong variable written and we solve for \(L\))

Subtract \(2W\) from both sides: \(P - 2W=2L\). Then divide both sides by 2: \(L=\frac{P - 2W}{2}\)

Step2: Solve \(-7n = 28\)

Divide both sides by \(-7\): \(n=\frac{28}{-7}=- 4\)

Step3: Solve for the dimensions of the rectangle

Let the width of the rectangle be \(w\) cm. Then the length \(l = w + 5\) cm. The perimeter formula for a rectangle is \(P=2(l + w)\). Given \(P = 48\) cm, so \(48=2((w + 5)+w)\). First, simplify the right - hand side: \(48=2(2w + 5)=4w+10\). Subtract 10 from both sides: \(48-10 = 4w\), so \(38 = 4w\). Then \(w=\frac{38}{4}=9.5\) cm. And \(l=w + 5=9.5 + 5 = 14.5\) cm

Step4: Solve \(-3(x - 1)+2(x + 5)=x + 7\)

Expand the left - hand side: \(-3x+3 + 2x + 10=x + 7\). Combine like terms: \(-x+13=x + 7\). Add \(x\) to both sides: \(13 = 2x+7\). Subtract 7 from both sides: \(6 = 2x\). Divide both sides by 2: \(x = 3\)

Step5: Solve \(5\div\frac{1}{5}\)

Remember that dividing by a fraction is the same as multiplying by its reciprocal. So \(5\div\frac{1}{5}=5\times5 = 25\)

Step6: Determine which equations have infinitely many solutions

For \(2(x - 3)=2x-6\), expand the left - hand side: \(2x-6=2x - 6\), which is an identity and has infinitely many solutions.
For \(3x + 2=3x-2\), subtract \(3x\) from both sides: \(2=-2\), which is a contradiction and has no solutions.
For \(-4(x + 5)=-4x-20\), expand the left - hand side: \(-4x-20=-4x - 20\), which is an identity and has infinitely many solutions.
For \(\frac{x}{2}=\frac{x + 4}{2}+2\), multiply both sides by 2: \(x=x + 4+4\), subtract \(x\) from both sides: \(0 = 8\), which is a contradiction and has no solutions. The equations with infinitely many solutions are \(2(x - 3)=2x-6\) and \(-4(x + 5)=-4x-20\)

Step7: Solve \((48 - 18)\div(15 - 3)\)

First, calculate the numerator and denominator: \(48-18 = 30\) and \(15 - 3=12\). Then \(\frac{30}{12}=\frac{5}{2}=2.5\)

Answer:

1. \(L=\frac{P - 2W}{2}\)
2. \(n=-4\)
3. Width \(=9.5\) cm, Length \(=14.5\) cm
4. \(x = 3\)
5. \(25\)
6. a. \(2(x - 3)=2x-6\), c. \(-4(x + 5)=-4x-20\)
7. \(2.5\)