Classifying Rectangles and Solving a System of Linear Equations

Math Problem Statement

Identify which of the shapes can be classified as a rectangle. Also, solve the following system of equations: 2x + y = 5 and -3x + 2y = 4.

The image contains a geometry problem and a system of linear equations problem. I'll break down each part and provide solutions.

The task is to identify which of the shapes can be classified as a rectangle. A rectangle has four right angles.

The shapes provided are:

Solution:

Shape 2 and Shape 4 can be classified as rectangles because they have four right angles.

The problem asks to solve the following system of equations by substitution or elimination.

\text{Equation 1:} & \quad 2x + y = 5 \\ \text{Equation 2:} & \quad -3x + 2y = 4 \end{align*}$$ #### **Step-by-Step Solution:** **Step 1: Solve for $$y$$ in Equation 1.** $$y = 5 - 2x$$ **Step 2: Substitute $$y = 5 - 2x$$ into Equation 2.** $$-3x + 2(5 - 2x) = 4$$ **Step 3: Expand and solve for $$x$$.** $$-3x + 10 - 4x = 4$$ $$-7x + 10 = 4$$ $$-7x = 4 - 10$$ $$-7x = -6$$ $$x = \frac{6}{7}$$ **Step 4: Substitute $$x = \frac{6}{7}$$ back into $$y = 5 - 2x$$.** $$y = 5 - 2\left(\frac{6}{7}\right)$$ $$y = 5 - \frac{12}{7}$$ $$y = \frac{35}{7} - \frac{12}{7}$$ $$y = \frac{23}{7}$$ **Solution:** The solution to the system of equations is: $$x = \frac{6}{7}, \quad y = \frac{23}{7}$$ ### Do you have any questions or would you like further details? Here are some related questions that might help: 1. How can you classify different types of quadrilaterals based on their properties? 2. What other methods can be used to solve a system of equations? 3. How do you verify that a solution to a system of equations is correct? 4. Can the solution to a system of equations be represented graphically? How? 5. What are the implications of having no solution or infinitely many solutions in a system of equations? **Tip:** When solving systems of equations, always double-check your substitution or elimination steps to avoid simple errors.

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Geometry
Algebra
Systems of Equations

Slope formula
Substitution method for solving systems of equations
Linear equations in two variables

Properties of rectangles
Fundamental theorem of algebra related to solving linear equations

Grades 7-9

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