Question

solve multi - step linear equations

1. lee is planting a rectangular section of grass.

if the perimeter of the rectangle is 96 feet, what are the length and width of the rectangular section?

1. math on the spot solve each equation

a. 3n + 1 = 19
b. 21=-2p - 5

1. reason what step would you perform first to solve the following equation? explain your reasoning.

\\(\frac{1}{4}(12 - 8x)=\frac{2}{3}(6x)\\)

1. heather has a family phone plan. the monthly payment for the family plan includes a $70 charge for unlimited talk and text, a $20 line fee per phone and a $22.91 equipment fee for each phone. their total monthly bill is $241.64. write and solve an equation to find how many phones are on the plan.

solve each equation. check your solution

1. a - 3(a - 1)=3(2 + 1)
2. 5y - 3(2 - y)=10
3. 1.2x - 2 = 7+0.9x
4. -k + 4(k + 1)=2k
5. 4(\frac{x}{6}+5)=2x + 10
6. 3w+\frac{w}{2}+1 = 10 - w

Explanation:

Step1: Solve problem 1

Let the length of the rectangle be \(l = 6x - 15\) and width be \(w=x\). The perimeter formula for a rectangle is \(P = 2(l + w)\). Given \(P=96\), so \(96=2((6x - 15)+x)\). First, simplify the equation inside the parentheses: \(96 = 2(7x-15)\). Then distribute the 2: \(96=14x - 30\). Add 30 to both sides: \(96 + 30=14x\), so \(126 = 14x\). Divide both sides by 14: \(x = 9\). The length \(l=6x - 15=6\times9-15=54 - 15 = 39\) feet and the width \(w = 9\) feet.

Step2: Solve problem 2A

For the equation \(3n+1 = 19\), subtract 1 from both sides: \(3n=19 - 1=18\). Then divide both sides by 3: \(n = 6\).

Step2: Solve problem 2B

For the equation \(21=-2p - 5\), add 5 to both sides: \(21 + 5=-2p\), so \(26=-2p\). Divide both sides by - 2: \(p=-13\).

Step3: Solve problem 3

For the equation \(\frac{1}{4}(12 - 8x)=\frac{2}{3}(6x)\), first simplify the left - hand side: \(\frac{1}{4}\times12-\frac{1}{4}\times8x = 3 - 2x\), and the right - hand side is \(4x\). So the equation becomes \(3-2x = 4x\). Add \(2x\) to both sides to get all \(x\) terms on one side.

Step4: Solve problem 4

Let the number of phones be \(x\). The monthly bill equation is \(70+(20 + 22.91)x=241.64\). First, simplify the left - hand side: \(70 + 42.91x=241.64\). Subtract 70 from both sides: \(42.91x=241.64 - 70 = 171.64\). Divide both sides by 42.91: \(x = 4\).

Step5: Solve problem 5

For the equation \(a-3(a - 1)=3(2 + 1)\), first distribute on both sides. Left - hand side: \(a-3a + 3\), right - hand side: \(9\). So the equation is \(a-3a + 3=9\). Combine like terms: \(-2a+3 = 9\). Subtract 3 from both sides: \(-2a=6\). Divide by - 2: \(a=-3\).

Step6: Solve problem 6

For the equation \(5y-3(2 - y)=10\), distribute the - 3: \(5y-6 + 3y=10\). Combine like terms: \(8y-6 = 10\). Add 6 to both sides: \(8y=16\). Divide by 8: \(y = 2\).

Step7: Solve problem 7

For the equation \(1.2r-2=7 + 0.9r\), subtract \(0.9r\) from both sides: \(1.2r-0.9r-2=7\), so \(0.3r-2 = 7\). Add 2 to both sides: \(0.3r=9\). Divide by 0.3: \(r = 30\).

Step8: Solve problem 8

For the equation \(-k + 4(k + 1)=2k\), distribute the 4: \(-k+4k + 4=2k\). Combine like terms: \(3k + 4=2k\). Subtract \(3k\) from both sides: \(4=-k\), so \(k=-4\).

Step9: Solve problem 9

For the equation \(4(\frac{x}{6}+5)=2x + 10\), distribute the 4: \(\frac{4x}{6}+20=2x + 10\). Simplify \(\frac{4x}{6}=\frac{2x}{3}\). Subtract \(\frac{2x}{3}\) from both sides and subtract 10 from both sides: \(20 - 10=2x-\frac{2x}{3}\), \(10=\frac{6x - 2x}{3}=\frac{4x}{3}\). Multiply both sides by \(\frac{3}{4}\): \(x=\frac{30}{4}=7.5\).

Step10: Solve problem 10

For the equation \(3w+\frac{w}{2}+1 = 10 - w\), first get a common denominator for the \(w\) terms. Multiply through by 2 to clear the fraction: \(6w+w + 2=20 - 2w\). Combine like terms: \(7w+2=20 - 2w\). Add \(2w\) to both sides: \(9w+2=20\). Subtract 2 from both sides: \(9w=18\). Divide by 9: \(w = 2\).

Answer:

1. Length: 39 feet, Width: 9 feet

2A. \(n = 6\)
2B. \(p=-13\)

1. First, simplify both sides and then get all \(x\) terms on one side.
2. 4 phones
3. \(a=-3\)
4. \(y = 2\)
5. \(r = 30\)
6. \(k=-4\)
7. \(x = 7.5\)
8. \(w = 2\)