Solve The Following Equations: 1. X + 3(2x - 1) = 4(2 - X) 2. (5/6)x - (1/3) = 12 - (2/5)x People Are Setting Up Benches For A Student Meeting. There Are A Certain Number Of Students, And If They Sit 5 Per Bench, 4 Students Are Left Without A Seat. If They Sit 7 Per Bench, One Bench Is Left Completely Empty. Determine The Number Of Students And The Number Of Benches.

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This article dives into solving two distinct mathematical problems. The first part focuses on solving linear equations, a fundamental concept in algebra. We'll walk through the steps to solve two specific equations, demonstrating how to isolate the variable and find its value. The second part presents a word problem involving students and benches, requiring us to translate the given information into equations and solve for the unknowns. This problem highlights the application of linear equations in real-world scenarios.

1. Solving Linear Equations

Linear equations are algebraic equations where the highest power of the variable is 1. Solving a linear equation involves finding the value of the variable that makes the equation true. This is typically achieved by performing algebraic operations on both sides of the equation to isolate the variable.

1.1 Equation 1: x + 3(2x - 1) = 4(2 - x)

To solve the equation x + 3(2x - 1) = 4(2 - x), we will follow a series of steps to isolate the variable 'x'. Let's break down the solution:

  1. Distribute: The first step involves distributing the constants outside the parentheses to the terms inside. This means multiplying 3 by both terms inside the first parenthesis (2x - 1) and multiplying 4 by both terms inside the second parenthesis (2 - x). This gives us:

    x + 6x - 3 = 8 - 4x

    This step is crucial because it simplifies the equation by removing the parentheses, making it easier to combine like terms.

  2. Combine Like Terms: Next, we need to combine the like terms on each side of the equation. On the left side, we have x and 6x, which can be combined to give 7x. So, the equation becomes:

    7x - 3 = 8 - 4x

    Combining like terms simplifies the equation further, bringing us closer to isolating the variable x.

  3. Move Variables to One Side: To isolate the variable x, we need to move all terms containing x to one side of the equation. We can do this by adding 4x to both sides of the equation. This eliminates the -4x term on the right side and adds it to the left side:

    7x + 4x - 3 = 8 - 4x + 4x

    Which simplifies to:

    11x - 3 = 8

    This step is a key part of the isolation process, centralizing all terms with x on one side.

  4. Move Constants to the Other Side: Now, we need to move all the constant terms (numbers without variables) to the other side of the equation. We can do this by adding 3 to both sides of the equation. This eliminates the -3 on the left side and adds it to the right side:

    11x - 3 + 3 = 8 + 3

    Which simplifies to:

    11x = 11

    This step ensures that only the term with x remains on one side, preparing us for the final isolation.

  5. Isolate the Variable: Finally, to isolate x, we need to divide both sides of the equation by the coefficient of x, which is 11. This will give us the value of x:

    11x / 11 = 11 / 11

    Which simplifies to:

    x = 1

    Thus, the solution to the equation x + 3(2x - 1) = 4(2 - x) is x = 1. This final step completes the process of solving for x, providing us with the numerical value that satisfies the original equation.

1.2 Equation 2: (5/6)x - (1/3) = 12 - (2/5)x

Now, let's solve the second linear equation: (5/6)x - (1/3) = 12 - (2/5)x. This equation involves fractions, so we'll need to handle them carefully. Here’s a step-by-step breakdown:

  1. Eliminate Fractions: The first step is to eliminate the fractions to simplify the equation. We can do this by finding the least common multiple (LCM) of the denominators, which are 6, 3, and 5. The LCM of 6, 3, and 5 is 30. Multiply every term in the equation by 30:

    30 * (5/6)x - 30 * (1/3) = 30 * 12 - 30 * (2/5)x

    This simplifies to:

    25x - 10 = 360 - 12x

    Eliminating fractions makes the equation easier to work with, as we now have integer coefficients.

  2. Move Variables to One Side: Next, we need to get all the terms containing x on one side of the equation. We can do this by adding 12x to both sides:

    25x + 12x - 10 = 360 - 12x + 12x

    This simplifies to:

    37x - 10 = 360

    Moving variables to one side centralizes the x terms, bringing us closer to isolating x.

  3. Move Constants to the Other Side: Now, we need to move the constant term to the other side of the equation. We can do this by adding 10 to both sides:

    37x - 10 + 10 = 360 + 10

    This simplifies to:

    37x = 370

    Moving constants to one side isolates the variable term, preparing us for the final step.

  4. Isolate the Variable: Finally, we isolate x by dividing both sides of the equation by the coefficient of x, which is 37:

    37x / 37 = 370 / 37

    This simplifies to:

    x = 10

    Therefore, the solution to the equation (5/6)x - (1/3) = 12 - (2/5)x is x = 10. This final step gives us the value of x that satisfies the original equation with fractions.

2. Bench Seating Problem

Word problems often require translating real-world scenarios into mathematical equations. This problem involves students and benches, and we need to find the number of students and benches.

2.1 Problem Setup

The problem states that if students sit 5 per bench, 4 students are left without a seat. If they sit 7 per bench, one bench is left empty. This gives us two scenarios that we can translate into equations. To solve this problem, we can use a system of linear equations. Let's define our variables:

  • Let s represent the number of students.
  • Let b represent the number of benches.

Now, we can translate the two scenarios into equations:

  • Scenario 1: If students sit 5 per bench, 4 students are left without a seat. This means the total number of students is 5 times the number of benches, plus 4. So, the equation is:

    s = 5b + 4

    This equation represents the relationship between the number of students and benches when students are seated 5 per bench, with 4 students left standing.

  • Scenario 2: If they sit 7 per bench, one bench is left empty. This means the number of students is 7 times the number of benches minus 1 (since one bench is empty). So, the equation is:

    s = 7(b - 1)

    This equation represents the relationship when students are seated 7 per bench, with one bench remaining unoccupied.

2.2 Solving the System of Equations

We now have a system of two equations with two variables:

  1. s = 5b + 4
  2. s = 7(b - 1)

We can solve this system using substitution or elimination. Here, we'll use the substitution method since both equations are already solved for s.

Since both equations are equal to s, we can set them equal to each other:

5b + 4 = 7(b - 1)

Now, let’s solve for b:

  1. Distribute: First, distribute the 7 on the right side of the equation:

    5b + 4 = 7b - 7

    This step expands the equation, making it easier to combine like terms.

  2. Move Variables to One Side: Next, we want to get all the terms with b on one side. Subtract 5b from both sides:

    5b - 5b + 4 = 7b - 5b - 7

    This simplifies to:

    4 = 2b - 7

    Moving variables to one side isolates the b terms.

  3. Move Constants to the Other Side: Now, add 7 to both sides to isolate the term with b:

    4 + 7 = 2b - 7 + 7

    This simplifies to:

    11 = 2b

    Moving constants to the other side prepares us for solving for b.

  4. Isolate the Variable: Finally, divide both sides by 2 to solve for b:

    11 / 2 = 2b / 2

    This gives us:

    b = 5.5

    However, since the number of benches must be a whole number, there seems to be an error in the problem statement or our interpretation. Let's re-evaluate the second scenario.

    The second scenario states that if students sit 7 per bench, one bench is left empty. This means that all students are seated on (b - 1) benches. Thus, the correct equation for the second scenario should be:

    s = 7(b - 1)

    We proceed by setting the two equations equal to each other:

    5b + 4 = 7(b - 1)

    Distribute the 7:

    5b + 4 = 7b - 7

    Subtract 5b from both sides:

    4 = 2b - 7

    Add 7 to both sides:

    11 = 2b

    Divide by 2:

    b = 11 / 2 = 5.5

    There is still an issue because benches must be a whole number. Let’s reconsider the interpretation of