A rectangle has a length of \( 3x + 2 \) and a width of \( x - 1 \). If \( x = 5 \), what is the area of the rectangle?
["Title: How to Calculate the Area of a Rectangle with Algebraic Dimensions (4x+1 Example)", "Meta Description: Learn how to calculate the area of a rectangle when its length and width are given algebraically. Our example uses ( \ ext{length} = 3x + 2 ), ( \ ext{width} = x - 1 ), and evaluates at ( x = 5 ).", "---", "### Introduction", "Geometry isn’t just for simple shapes—rectangles with algebraic dimensions are a common way to model real-world problems in algebra and advanced math. Understanding how to calculate the area using variables and substitute specific values is essential for anyone studying high school or college math.", "In this article, we explore how to find the area of a rectangle when the length and width are expressions in terms of a variable ( x ). We’ll walk through the example: if the length is ( 3x + 2 ), the width is ( x - 1 ), and ( x = 5 ), what is the area?", "---", "### Step 1: Write Down the Expressions", "We are given:", "- Length ( = 3x + 2 )\n- Width ( = x - 1 )\n- Value of ( x = 5 )", "---", "### Step 2: Substitute ( x = 5 ) into Each Expression", "Substituting ( x = 5 ) gives:", "- Length ( = 3(5) + 2 = 15 + 2 = 17 )\n- Width ( = 5 - 1 = 4 )", "So, the rectangle has:\n- Length = 17 units\n- Width = 4 units", "---", "### Step 3: Use the Area Formula", "The area ( A ) of a rectangle is calculated by multiplying length by width:", "[\nA = \ ext{length} \ imes \ ext{width}\n]", "Substitute the values:", "[\nA = 17 \ imes 4\n]", "[\nA = 68\n]", "---", "### Step 4: Alternative: Use Algebra First, Then Substitute", "Alternatively, you can multiply the expressions first and then substitute:", "[\n\ ext{Area} = (3x + 2)(x - 1)\n]", "Expand the product using the distributive property (FOIL):", "[\n(3x)(x) + (3x)(-1) + (2)(x) + (2)(-1) = 3x^2 - 3x + 2x - 2 = 3x^2 - x - 2\n]", "Now substitute ( x = 5 ):", "[\n3(5)^2 - 5 - 2 = 3(25) - 5 - 2 = 75 - 5 - 2 = 68\n]", "Same result—verified!", "---", "### Conclusion", "When dealing with rectangles whose dimensions are expressed algebraically, always:", "1. Substitute the given variable value.\n2. Multiply the resulting length and width either directly or by expanding fully.\n3. Simplify the expression to find the exact area.", "In this case, with ( x = 5 ), the rectangle has dimensions 17 and 4, resulting in a clear and precise area of 68 square units.", "---", "### Key Takeaways:", "- Compound areas with algebraic sides simplify through substitution or expansion.\n- Always clarify whether expressions are pre-substituted or expanded.\n- This method applies broadly to any rectangular geometry with variable dimensions.", "---", "### Interested in More?\nCheck out our guides on solving algebra problems step-by-step, quadratic equations, and real-world applications of geometry in calculus!", "---\nKeywords: rectangle area formula, algebra geometry, calculating area with variables, solve for x in rectangle area, 3x+2 width 5, algebra rectangle problem, solving with x=5"]
