Math Problem Statement

Write and solve a system of linear equations to find the values of x and y, based on the trapezoid diagram.

Solution

The image contains a diagram that seems to represent a trapezoid with sides labeled in terms of xx and yy. Specifically, the sides are labeled as follows:

  • One diagonal side is labeled 3x3x,
  • Another diagonal side is labeled 7y7y,
  • The horizontal base is labeled 4x4x,
  • The top side is labeled 5y+55y + 5.

The problem asks to form and solve a system of linear equations to find the values of xx and yy.

Step 1: Identify relationships

From the trapezoid, the sides are likely equal, so we can form two equations based on equal lengths:

  1. Set the lengths of the two diagonal sides equal: 3x=7y(equation 1)3x = 7y \quad \text{(equation 1)}
  2. Set the lengths of the top and bottom sides equal: 4x=5y+5(equation 2)4x = 5y + 5 \quad \text{(equation 2)}

Step 2: Solve the system of equations

Solve equation (1) for xx:

x=7y3x = \frac{7y}{3}

Substitute this into equation (2):

4(7y3)=5y+54\left(\frac{7y}{3}\right) = 5y + 5 Simplifying: 28y3=5y+5\frac{28y}{3} = 5y + 5 Multiply through by 3 to eliminate the fraction: 28y=15y+1528y = 15y + 15 Simplifying: 28y15y=1528y - 15y = 15 13y=1513y = 15 y=1513y = \frac{15}{13}

Substitute y=1513y = \frac{15}{13} into equation (1):

3x=7(1513)3x = 7\left(\frac{15}{13}\right) x=7×153×13=10539=3513x = \frac{7 \times 15}{3 \times 13} = \frac{105}{39} = \frac{35}{13}

Final Solution:

x=3513,y=1513x = \frac{35}{13}, \quad y = \frac{15}{13}

Would you like further explanation or clarification on any steps?

Here are 5 related questions you could explore:

  1. How can this system of linear equations be solved using a different method, like graphing?
  2. What is the geometric meaning of the lengths in a trapezoid?
  3. How can you verify the solution using substitution into the original equations?
  4. What is the significance of the slope of the diagonal sides in this context?
  5. How does solving this system algebraically compare to solving it by elimination?

Tip: When solving systems of linear equations, it's often helpful to first solve one equation for a variable, then substitute into the second equation.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Algebra

Formulas

3x = 7y
4x = 5y + 5

Theorems

Substitution Method for Solving Systems of Equations

Suitable Grade Level

Grades 8-10