Textbook Question
In Exercises 1–18, solve each system by the substitution method. x^2+y^2=25, x-y=1
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7. Systems of Equations & Matrices
Two Variable Systems of Linear Equations
6:32 minutesProblem 36
Textbook Question
The perimeter of a rectangle is 26 meters and its area is 40 square meters. Find its dimensions.
Verified step by step guidance1
Step 1: Recall the formulas for the perimeter and area of a rectangle. The perimeter is given by \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width. The area is given by \( A = l \cdot w \).
Step 2: Substitute the given values into the formulas. For the perimeter, \( 26 = 2(l + w) \). For the area, \( 40 = l \cdot w \).
Step 3: Solve the perimeter equation for one variable, such as \( w \). Divide both sides of \( 26 = 2(l + w) \) by 2 to get \( l + w = 13 \). Then solve for \( w \) to get \( w = 13 - l \).
Step 4: Substitute \( w = 13 - l \) into the area equation \( 40 = l \cdot w \). This gives \( 40 = l \cdot (13 - l) \). Expand the equation to get \( 40 = 13l - l^2 \). Rearrange into standard quadratic form: \( l^2 - 13l + 40 = 0 \).
Step 5: Solve the quadratic equation \( l^2 - 13l + 40 = 0 \) using factoring, completing the square, or the quadratic formula \( l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -13 \), and \( c = 40 \). Once \( l \) is found, substitute back into \( w = 13 - l \) to find \( w \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Perimeter of a Rectangle
The perimeter of a rectangle is the total distance around the rectangle, calculated by the formula P = 2(l + w), where l is the length and w is the width. In this problem, the perimeter is given as 26 meters, which provides a relationship between the length and width that can be used to find their values.
Area of a Rectangle
The area of a rectangle is the amount of space enclosed within its sides, calculated using the formula A = l × w. Here, the area is specified as 40 square meters, which creates another equation involving the length and width. Solving these equations simultaneously will yield the dimensions of the rectangle.
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Simultaneous Equations
Simultaneous equations are a set of equations with multiple variables that are solved together to find a common solution. In this case, the two equations derived from the perimeter and area of the rectangle can be solved simultaneously to determine the values of length and width, allowing us to find the rectangle's dimensions.
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