91Ó°ÊÓ

When solving a problem involving a system of equations, it is crucial to begin by identifying the variables. Variables are symbols or letters that represent unknown values that we aim to find. They are placeholders that allow us to create mathematical expressions, which we can then solve. In the given exercise, two different coins are involved: dimes and quarters. Therefore, we assign a variable to each.

• Dimes: Let the variable \( d \) represent the number of dimes in the bag.
• Quarters: Let the variable \( q \) represent the number of quarters.

Choosing suitable variables is the foundational step because it sets the stage for forming equations. Once we have our variables, we can move on to establish relationships between them.

An equation is a mathematical statement indicating that two expressions are equal. In the context of a system of equations, we create equations to reflect the relationships between our variables.
In the exercise, we have a total of 30 coins made up of dimes and quarters. This relationship is captured in the first equation:

• Total coins equation: \( d + q = 30 \)

The next relationship equates to the total value of these coins being \( \\(3.45 \). Dimes are worth \( \\)0.10 \) each, and quarters are worth \( \$0.25 \) each. Thus, the second equation relating to the total value of the coins is:

• Total value equation: \( 0.10d + 0.25q = 3.45 \)

These equations form a system that we need to solve to find the values of \( d \) and \( q \).

To solve a system of equations means to find the values of the variables that make all equations true simultaneously. There are various methods for solving systems of equations, such as substitution, elimination, or graphing. In this exercise, we use the substitution method.
Start with the total coins equation: \( d + q = 30 \). Solve for one variable in terms of the other; for instance, express \( d \) in terms of \( q \):

• \( d = 30 - q \)

Next, substitute this expression for \( d \) into the total value equation:\[ 0.10(30 - q) + 0.25q = 3.45 \] Simplify and solve the equation for \( q \):

• \( 3.00 - 0.10q + 0.25q = 3.45 \)
• \( 0.15q = 0.45 \)
• \( q = 3 \)

Now that you have \( q = 3 \), substitute back to find \( d \):

• \( d = 30 - 3 = 27 \)

Hence, solving shows there are 3 quarters and 27 dimes.

The concept of total value is essential in word problems dealing with financial scenarios. Total value refers to the combined worth of all items in a given problem, in our case, the coins in the bag. The total value of the coins is calculated in terms of their denominations and quantities.
For this exercise, each coin carries a specific value:

• Dimes: \( \\(0.10 \) each
• Quarters: \( \\)0.25 \) each

To find the overall total value, we multiply the number of each coin by its individual worth and then add these to give the combined total. Mathematically, this is expressed as the second equation:

• \( 0.10d + 0.25q = 3.45 \)

This equation signifies that when the individual values of the dimes and quarters are added up, they equal \( \$3.45 \). Understanding this concept allows us to form equations that correctly describe the situation, bridging the gap to solving for unknown quantities.