Problem 83 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 2 x+y=10 \\ -x+y=-5 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = 5\) and \(y = 0\).

Step 1: Solve one equation for one variable

Choose one of the equations to solve for one of the variables. Here, solve the second equation for variable.\(-x + y = -5\).\(y = x -5\)

Step 2: Substitute the result into the other equation

Next, substitute the expression for into the other equation. The other equation is: \(2x + y = 10\). Rewrite it as: \(2x + (x - 5) = 10\)

Step 3: Simplify and solve for

Now, combine like terms and solve for : \(2x + x - 5 = 10\). This simplifies to: \(3x - 5 = 10\). Add 5 to both sides: \(3x = 15\). Divide both sides by 3: \(x = 5\)

Step 4: Substitute back to solve for

We now know that \(x = 5\). Substitute this value back into the equation \(y = x - 5\):\(y = 5 - 5\). Thus, \(y = 0\)

Step 5: Write the solution

The solution to the system of equations is \(x = 5\) and \(y = 0\)

Key Concepts

Solving systems of equations

A system of equations is a set of two or more equations with the same variables. Solving these systems means finding the values of the variables that make both equations true simultaneously. There are different methods to solve systems of equations: graphing, substitution, and elimination. This exercise focuses on the **substitution method**, where one equation is solved for one variable, and that expression is substituted into the other equation.
By mastering this method, you can efficiently solve systems and understand how variables interact in multiple equations.

Algebraic substitution

The substitution method is a straightforward way to solve systems of equations. Here is the process:

• **Step 1**: Solve one of the equations for one variable. In our example, we start by solving the second equation \( -x + y = -5 \) for \( y \), resulting in \( y = x - 5 \).
• **Step 2**: Substitute this expression into the other equation. We replace \( y \) in \( 2x + y = 10 \) with \( x - 5 \), giving us \( 2x + (x - 5) = 10 \).
• **Step 3**: Simplify and solve for the remaining variable. Combine like terms to get \( 3x - 5 = 10 \), solve for \( x \) to find \( x = 5 \).
• **Step 4**: Substitute back to find the other variable. Using \( y = x - 5 \) and substituting \( x = 5 \), we get \( y = 0 \).
• **Step 5**: Write the solution. The solution to the system is \( x = 5 \) and \( y = 0 \).

Understanding how to use substitution will make solving systems of equations more intuitive.

Elementary algebra exercises

Elementary algebra exercises, like the one in this example, help build foundational skills in mathematics. These exercises often involve:

• **Combining Like Terms**: Simplifying expressions by adding or subtracting coefficients of similar variables.
• **Balancing Equations**: Ensuring both sides of an equation have equal value, often by adding, subtracting, multiplying, or dividing terms.
• **Isolating Variables**: Rearranging an equation to get one variable alone on one side of the equation.
• **Checking Solutions**: Substituting your solutions back into the original equations to verify correctness.

Practicing these steps in a structured exercise helps in enhancing problem-solving skills required for more complex algebra and beyond.

Step-by-step solutions

Providing a clear, step-by-step solution is crucial for understanding and learning how to solve mathematical problems. Breaking down each problem into smaller, manageable steps helps minimize errors and builds confidence. Here is how a detailed step-by-step approach works:

• **Identify the Problem**: Clearly state what needs to be solved.
• **Plan the Strategy**: Decide on the method or strategy to use, such as the substitution method.
• **Execute the Plan**: Solve the problem step by step as outlined. For example, solve the first equation for one variable, substitute into the second equation, and solve.
• **Review the Solution**: Check each step and verify the solution by plugging variables back into the original equations.

This systematic approach not only helps in tackling algebra problems but also develops critical thinking and logical reasoning skills.