Problem 60 - College Algebra

Question

Use a system of linear equations to solve Exercises \(57-67\) The graph shows the calories in some favorite fast foods. Use the information in Exercises \(57-58\) to find the exact caloric content of the specified foods. (GRAPH CAN'T COPY) Two medium eggs and three cups of ice cream contain 701 milligrams of cholesterol. One medium egg and one cup of ice cream exceed the suggested daily cholesterol intake of 300 milligrams by 25 milligrams. Determine the cholesterol content in each item.

Step-by-Step Solution

Answer

One medium egg contains 274 milligrams of cholesterol and one cup of ice cream contains 51 milligrams of cholesterol.

Step 1: Translate the Problem into Equations

Let's denote the cholesterol content in a medium egg by \(E\) and in a cup of ice cream by \(I\). From the information, you can write the following equations: The first piece of information translates to \(2E + 3I = 701\), and the second to \(E + I = 300 + 25 = 325\).

Step 2: Solve the System of Equations

This system of equations can be solved using substitution or elimination method. In this case, the second equation \(E + I = 325\) can be transformed into \(E = 325 - I\). Substituting this into the first equation gives \(2(325-I) + 3I = 701\), or \(650 - 2I + 3I = 701\). Combining like terms leads to \(I = 51\). Substituting \(I = 51\) into \(E + I = 325\) gives \(E = 325 - 51 = 274\).

Step 3: Interpret the Result

The solutions \(E = 274\) and \(I = 51\), mean that one medium egg contains 274 milligrams of cholesterol and one cup of ice cream contains 51 milligrams of cholesterol.

Key Concepts

cholesterol content

Understanding cholesterol content is crucial for making informed dietary choices. Cholesterol is a fat-like substance found in your body and in foods. It is essential for building cells, but too much can be harmful. In this exercise, you need to determine how much cholesterol is in a medium egg and a cup of ice cream. You're given two relationships involving these quantities and are asked to find their exact values. Knowing the cholesterol content helps you manage intake, either meeting or staying below recommended daily values.

elimination method

The elimination method is a powerful tool for solving systems of linear equations. It involves manipulating the equations to cancel out one of the variables, making it easier to solve for the remaining one. In this exercise, we have two equations:

• Equation 1: \( 2E + 3I = 701 \)
• Equation 2: \( E + I = 325 \)

To use the elimination method, you first manipulate one of the equations to allow a variable to cancel out when added or subtracted from the other equation. This transforms the system into one equation with one variable, which can then be solved straightforwardly. It is a systematic process that ensures accuracy when dealing with more complex problems. Here, we focus on isolating 'I' by properly substituting 'E', making it simple to find 'I' and subsequently 'E'.

translation of word problems

Translating word problems into mathematical equations is the first critical step in solving these types of exercises. Word problems often describe real-world situations and relationships in words, which must be accurately converted into algebraic expressions. In this exercise,

• the statement about the cholesterol in eggs and ice cream is converted into equations representing the quantities mathematically.
• It's important to carefully interpret the information given. For instance, 'exceeding by 25 milligrams' involves performing an addition operation: \(E + I = 325\).

This kind of translation requires attention to detail to ensure that all relationships and operations are correctly represented. Proper translation sets the foundation for solving the problem effectively.

solving algebraic equations

Solving algebraic equations involves finding the value of unknown variables that make the equation true. In this exercise, the system of equations provides a clear path to finding the specific cholesterol contents:

• First, substitute \( E = 325 - I \) into the first equation from the step by step solution.
• This substitution technique simplifies the system to a single equation: \( 650 - 2I + 3I = 701 \), which can be combined to simplify further.
• Completing the calculations, we find \( I = 51 \) and, using another substitution step, find \( E = 274 \).

This systematic approach involves logical steps that transform the equations from complex to simple, allowing the values of 'E' and 'I' to emerge. By breaking down each step carefully, we reduce errors and misunderstandings, achieving accurate results.