Problem 13 - Algebra 1

Question

Use the graph to check the answer to the problem below. You have \(\$ 320\) and save \(\$ 10\) each week. Your brother has \(\$ 445\) and spends his income, plus \(\$ 15\) of his savings each week When will you and your brother have the same amount in savings? $$ \begin{aligned} 320+10 t &=445-15 t \\ 320 &=445-5 t \\ -125 &=-5 t \\ 25 &=t \quad \text { in } 25 \text { weeks } \end{aligned} $$

Step-by-Step Solution

Answer

The point in time (weeks) when you will have the same amount of savings as your brother is in 5 weeks. However, there seems to be a mistake in the solved exercise. The answer should be in 5 weeks, not 25.

Step 1: Set up the equation

The first step is to set up and solve the equation. The value of savings can be represented as \( 320 + 10t \) for you and \( 445 - 15t \) for your brother. The point at which you both will have saved the same amount is when \( 320 + 10t = 445 - 15t \). This equation sets the two amounts equal to each other to find the time \( t \) at which both amounts will match.

Step 2: Simplify the equation

Adding \( 15t \) to both sides of the equation will eliminate \( t \) from the right side, resulting in the equation \( 320 + 25t = 445 \). Next, subtract \( 320 \) from both sides, simplifying the equation to \( 25t = 125 \).

Step 3: Solve for t

Finally, the third step is to solve for \( t \). By dividing both sides of the equation by \( 25 \), we determine that \( t = 5 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They form the backbone of algebra, allowing us to represent real-world situations with mathematical symbols. In the scenario given, the expressions\[ 320 + 10t \] and\[ 445 - 15t \] are used to model savings over time for you and your brother, respectively.

• The first expression, \( 320 + 10t \), represents your savings. It starts with \( \\(320 \) and increases by \( \\)10 \) each week, noted by the variable \( t \).
• The second expression, \( 445 - 15t \), shows your brother's savings. He begins with \( \\(445 \) and loses \( \\)15 \) each week.

These expressions help set up a linear equation to find when you both will have the same amount saved. Each component of the expressions has a clear connection to real-world actions, such as earning or spending money over time.

Solving Equations

Solving equations is a fundamental aspect of algebra that consists of finding the value of variables that make the equation true. In our problem, we want to determine when you and your brother will have equal savings.To solve the equation \( 320 + 10t = 445 - 15t \), follow these steps: - **Eliminate the variable on one side**: Add \(15t\) to both sides to concentrate all terms with \(t\) on one side. This simplifies the equation to \(320 + 25t = 445\). - **Isolate the term with \(t\)**: Subtract \(320\) from both sides to further simplify it to \(25t = 125\). - **Solve for \(t\)**: Divide both sides by \(25\) to find \(t = 5\). This calculation reveals that it will take 25 weeks for the savings to equalize.Each mathematical operation systematically simplifies the equation, demonstrating the power of algebra to solve real-world problems.

Graphs in Algebra

Graphs in algebra are visual tools that help us understand the relationships between variables in equations. They can provide insights that might not be obvious from the algebraic expressions alone.For this problem, plotting the savings over time for both you and your brother can clearly show where the lines (representing each savings situation) intersect:

• Your line starts at \( \\(320 \) and rises by \( \\)10 \) each week, creating an upward slope.
• Your brother's line begins at \( \\(445 \) but drops by \( \\)15 \) weekly, forming a downward slope.

The point where these two lines cross on the graph is crucial, as it visually confirms the week when your and your brother's savings are equal. This intersection, which occurs at \( t = 25 \), verifies the solution found through solving the equation. Using graphs is an excellent method for checking answers and offers a visual confirmation of algebraic solutions.