Solving Multi-Step Equations in One Variable

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.

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.

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.