Question

1. solve each of these equations. explain or show your reasoning.$3(2x - 3) = 4x + 15LXB03(4 - 3z) = 6(z + 4)$4. in a game, the number of points scored are plotted against the number of correct answers.based on this information, is the points scored a linear function of the number of correct answers? explain your thinking.5. a carpenter designs a shed so that the floor has a perimeter of 48 feet with the width measuring one-half the length minus 4.a. using $l$ to represent the length of the shed floor and $w$ to represent its width, complete the system of equations below that describes this situation.b. what are the dimensions of the shed floor?

Explanation:

Problem 3

Step1: Expand left side

$3(2x-3)=6x-9$
Equation becomes: $6x-9=4x+15$

Step2: Isolate x terms

Subtract $4x$ from both sides: $6x-4x-9=15$
Simplify: $2x-9=15$

Step3: Isolate constant terms

Add 9 to both sides: $2x=15+9$
Simplify: $2x=24$

Step4: Solve for x

Divide by 2: $x=\frac{24}{2}=12$

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Step1: Isolate y terms

Add $8y$ and subtract 12: $-16-12=6y+8y$

Step2: Simplify both sides

$-28=14y$

Step3: Solve for y

Divide by 14: $y=\frac{-28}{14}=-2$

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Step1: Expand both sides

Left: $3(4-3z)=12-9z$; Right: $6(z+4)=6z+24$
Equation becomes: $12-9z=6z+24$

Step2: Isolate z terms

Subtract $6z$ and 12: $-9z-6z=24-12$
Simplify: $-15z=12$

Step3: Solve for z

Divide by -15: $z=\frac{12}{-15}=-\frac{4}{5}$

A linear function has a constant rate of change (slope) between all points. Identify the plotted points: (3,15), (6,35), (9,48). Calculate the slope between the first two points: $\frac{35-15}{6-3}=\frac{20}{3}\approx6.67$. Calculate the slope between the last two points: $\frac{48-35}{9-6}=\frac{13}{3}\approx4.33$. The slopes are not equal, so there is no constant rate of change.

Part a

Step1: Perimeter equation

Perimeter of rectangle: $2(l+w)=48$
Simplify: $l+w=24$

Step2: Width-length relationship

Width = $\frac{1}{2}$ length - 4: $w=\frac{1}{2}l-4$

Part b

Step1: Substitute w into perimeter eq

Substitute $w=\frac{1}{2}l-4$ into $l+w=24$:
$l+\frac{1}{2}l-4=24$

Step2: Simplify to solve for l

$\frac{3}{2}l=24+4=28$
$l=28\times\frac{2}{3}=\frac{56}{3}\approx18.67$

Step3: Solve for w

$w=\frac{1}{2}\times\frac{56}{3}-4=\frac{28}{3}-\frac{12}{3}=\frac{16}{3}\approx5.33$

Answer:

$x=12$
$y=-2$
$z=-\frac{4}{5}$

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Problem 4