Problem 75 - Algebra 1: Concepts and Skills

Question

Use the substitution method to solve the linear system. $$ \begin{aligned} &-x+y=0\\\ &2 x+y=0 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is x = 0, y = 0.

Step 1: Solve One Equation for One Variable

Choose one of the equations to solve for one variable. In this case, the first equation is simpler, so it will be used. Solving -x + y = 0 for y gives us \( y = x \).

Step 2: Substitute the Expression into the Other Equation

Now that we know \( y = x \), substitute this into the second equation. This gives us \( 2x + x = 0 \). Simplifying this gives us \( 3x = 0 \).

Step 3: Solve for the Variable

After substitution, solve the resulting equation for the variable. In this case, solving \( 3x = 0 \) for x gives us \( x = 0 \).

Step 4: Substitute x=0 Into the Initial Equation

Now substitute \( x = 0 \) into the initial equation, which is \( y = x \). So, \( y = 0 \)

Key Concepts

Linear Equations

Linear equations are a fundamental part of algebra, often represented as straight lines when graphed on a coordinate plane. These equations generally take the form:

• \( ax + by = c \)

where \( a \), \( b \), and \( c \) are constants. The term "linear" suggests that each term in the equation is either a constant or the product of a constant and a single variable.
Linear equations are crucial because they model real-world situations where relationships are proportional. For instance, you might encounter them in calculating costs, speed, or other measurable phenomena derived from a straight line relationship.
Understanding linear equations helps in analyzing data and predicting outcomes. Graphically, they are straightforward to interpret as they form straight lines, which makes visualization and interrelation of data easier.

Solving Systems of Equations

Solving systems of equations involves finding the values of variables that satisfy multiple equations simultaneously. A system of linear equations may have:

• One solution: where the lines intersect at a single point.
• No solutions: where the lines are parallel and never intersect.
• Infinite solutions: where the lines coincide completely, meaning they are the same line.

When using algebraic methods, such as substitution or elimination, solving systems of equations can reveal crucial values, such as equilibriums in economics or chemical reaction rates in science. Methods like graphing, substitution, and elimination help confirm or visualize potential solutions.
The process enhances problem-solving skills, involving logic and strategic thinking in balancing and rearranging equations to uncover hidden variable values.

Variable Substitution

Variable substitution is a method used to solve systems of equations, particularly those involving linear equations. The idea is to simplify one equation in terms of a single variable, then replace or "substitute" this expression into another equation to reduce the number of variables.
Consider the method steps shown in the original solution:

• First, solve one of the equations for one variable, often the easier one to manipulate. For instance, \(-x + y = 0\) can be transformed into \(y = x\).
• Substitute this expression into the other equation. Using our example, placing \(y = x\) in \(2x + y = 0\) simplifies it to \(3x = 0\).
• Solve this new equation for \(x\). In this case, you find \(x = 0\).
• Finally, substitute back to find \(y\). With \(x = 0\), we know \(y = x = 0\).

This technique is powerful for simplifying complex algebraic problems, reducing multiple-steps into clearer, more digestible parts.