Mastering Solving Multi Step Equations in Algebra

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To make the tower of a cardboard castle, Zosia needs to cut out a triangle and a rectangle, both with the same area, whose dimensions are shown in the figure.

A triangle whose base is x millimeters and whose height is 40 millimeters. A rectangle with sides 12.5 and x+18 millimeters each.

What is the area of the triangle?

The area of a triangle is half the product of its base and its height. Let's start by writing an expression that represents the area of the triangle that Zosia needs. Triangle's Area [0.15cm] 1/2x* 40 = 40x/2 We need to know the value of x in order to calculate the area of the triangle. To find it, let's also write an expression representing the area of the rectangle that Zosia needs. We will use the fact that the area of a rectangle is equal to the length multiplied by the width. Rectangle's Area 12.5(x+18) Since the triangle and the rectangle have the same area, we can equate the expressions so that we get an equation in terms of x. 40x/2 = 12.5(x+18) We can find the value of x by solving the equation. To start, let's multiply both sides of the equation by 2 to get rid of the fractions.

Next, we can distribute 25 to clear all the parentheses.

Finally, we will group the variable terms on the left-hand side. We can then apply inverse operations to isolate x.

The value of x is 30, which means that the base of the triangle is 30 millimeters long. Knowing this, we can find the area of the triangle.

The area of the triangle is 600 square millimeters. Since the triangle and the rectangle have the same area, we could also substitute x=30 into the expression representing the area of the rectangle — we would get the same answer, though.

Four consecutive even numbers add up to

92 .

What is the next even number?

To determine the next even number, we need to find the value of the four consecutive even numbers. Let's start by calling the smallest of the four even numbers x. \begin{gathered} \bm{1}^\text{st} \to {\color{#0000FF}{x}} \end{gathered} Since the four numbers are even and consecutive, the second even number is two units greater than x. This means that we can write it as x+2. \begin{aligned} \bm{1}^\text{st} &\to {\color{#0000FF}{x}} \\ \bm{2}^\text{nd} &\to {\color{#009600}{x+2}} \end{aligned} The third even number is two units greater than the second one. Then, the third number equals x+2+2, which is the same as x+4. \begin{aligned} \bm{1}^\text{st} &\to {\color{#0000FF}{x}} \\ \bm{2}^\text{nd} &\to {\color{#009600}{x+2}} \\ \bm{3}^\text{rd} &\to {\color{#FD9000}{x+4}} \end{aligned} Similarly, the fourth even number is two units greater than the third one. Therefore, the fourth number equals x+4+2, or x+6. \begin{aligned} \bm{1}^\text{st} &\to {\color{#0000FF}{x}} \\ \bm{2}^\text{nd} &\to {\color{#009600}{x+2}} \\ \bm{3}^\text{rd} &\to {\color{#FD9000}{x+4}} \\ \bm{4}^\text{th} &\to {\color{#A800DD}{x+6}} \end{aligned} Now that we have expressions for each of the four consecutive even numbers in terms of x, we can add them and equate the sum to 92. x + ( x+2) + ( x+4) + ( x+6) = 92 We can find the value of x by solving the equation. Note that the equation has no fractions and the variable terms are all grouped on the left-hand side. This means that we can simply combine like terms and isolate the variable through inverse operations to solve the equation.

The value of x is 20, so the smallest of the four consecutive even numbers is 20. Knowing this, we can determine the next three even numbers. \begin{aligned} \bm{1}^\text{st} &\to {\color{#0000FF}{20}} \\ \bm{2}^\text{nd} &\to {\color{#009600}{22}} \\ \bm{3}^\text{rd} &\to {\color{#FD9000}{24}} \\ \bm{4}^\text{th} &\to {\color{#A800DD}{26}} \end{aligned} We can conclude that the fifth even number is 26+2=28.

In a triangle

A B C,

the side

\overline{A B}

3

centimeters shorter than

\overline{B C},

and

\overline{A C}

is twice as long as

\overline{A B} .

If the perimeter of the triangle is

23

centimeters, what is the length of the longest side of the triangle?

To determine the length of the longest side of the triangle, we need to know its three side lengths. Let's try to write expressions representing the side lengths in terms of one variable. To start, we can call x the length of BC. BC → x We are told that AB is 3 centimeters shorter than BC, which allows us to write the length of AB in terms of x. ABis3centimeters shorter thanBC ⇓ AB → x-3 Also, we know that AC is twice as long as AB. This means that the length of AC is 2 times the length of AB. ACis twice as long asAB ⇓ AC → 2(x-3) Now we have the side lengths written in terms of only one variable. The perimeter of a triangle equals the sum of the three side lengths. We know that the perimeter of △ ABC is 23 centimeters. Therefore, the sum of the three expressions is equal to 23. x + x-3 + 2(x-3) = 23 Next, let's solve the equation for x. We will start by distributing 2 to clear all the parentheses and continue by combining like terms.

Finally, we isolate the variable on the left-hand side by applying inverse operations.

The value of x is 8. This means that BC is 8 centimeters long. Knowing this, the other two side lengths can be calculated. BC & → 8 cm AB & → 8-3 = 5 cm AC & → 2(8-3) = 10 cm As we can see, the longest side of △ ABC is AC, which is 10 centimeters long.

The following figure is made of four identical rectangles and one square whose area is half the area of a rectangle.

Four identical rectangles and a square in the middle all forming a square

If the area of the figure is

54

square inches, what is the area of one of the rectangles?

We are told that the figure is made of four identical rectangles and one square. Let x be the area of each rectangle. Since the area of the square is half the area of a rectangle, the area of the square is 12x. Rectangle's Area &→ x [0.3em] Square's Area &→ 1/2x There are four rectangles in the figure and the area of all of them together is 4x. We can find the total area of the figure by adding together the area of the four rectangles and the area of the square. We also know that this area is 54 square inches, so we will set this expression equal to 54. 4x + 1/2x = 54 Let's solve the equation to find the value of x. To start, we will multiply both sides of the equation by 2 to get rid of the fraction. Then, we will combine like terms and isolate the variable using inverse operations.

The area of each rectangle is 12 square inches. This means that the area of the square is 6 square inches.

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