Problem 63 - Algebra 2

Question

PREREQUISITE SKILL Solve each system of equations. $$ \begin{array}{l}{4 x+y=14} \\ {4 x-y=10}\end{array} $$

Step-by-Step Solution

Answer

The solution is \((3, 2)\).

Step 1: Write down the equations

Given the system of equations: \( 4x + y = 14 \) and \( 4x - y = 10 \).

Step 2: Apply the Elimination Method

Add the two equations together to eliminate the variable \( y \): \( (4x + y) + (4x - y) = 14 + 10 \). This simplifies to \( 8x = 24 \).

Step 3: Solve for x

From \( 8x = 24 \), divide both sides by 8 to solve for \( x \): \( x = \frac{24}{8} = 3 \).

Step 4: Substitute x into one of the original equations

Substitute \( x = 3 \) back into the first equation \( 4x + y = 14 \): \( 4(3) + y = 14 \).

Step 5: Solve for y

Simplify \( 4(3) + y = 14 \) to get \( 12 + y = 14 \). Subtract 12 from both sides to find \( y = 2 \).

Step 6: Write the solution as an ordered pair

The solution to the system of equations is \( (x, y) = (3, 2) \).

Key Concepts

Elimination Method

The elimination method is a powerful technique for solving systems of equations by removing one variable to make the equations easier to solve. In this method, we focus on aligning the equations so that adding or subtracting them will cancel out one of the variables, leaving us with a simpler equation with only one variable.
To use elimination effectively:

• Ensure one of the variables in both equations has the same or opposite coefficient.
• Add or subtract the equations to eliminate that variable.
• Solve the resulting single-variable equation.

This approach simplifies complex systems, allowing us to find the variable values step by step.

Solving for Variables

Solving systems of equations involves finding values for the variables that satisfy all given equations simultaneously. Once we have simplified the equations using methods like elimination or substitution, each solution represents the point where the equations intersect on a graph.
To solve for variables:

• First, simplify one of the equations, if needed, to express one variable in terms of the others.
• Use this expression in the other equation to find exact values.

The goal is through these steps, to get an accurate value for each variable so that they satisfy all equations in the system.

Substitution Method

The substitution method offers an alternative path to solving systems of equations. Instead of keeping both equations in terms of two variables, this method focuses on solving one equation for one variable, then substituting that expression into the other equation.
Here's how to approach it:

• Choose one equation and solve it for one variable.
• Substitute this expression into the other equation.
• Solve the new equation for the remaining variable.
• Substitute back to find the other variable.

The substitution method can be particularly useful when one equation is already simplified or when the coefficients do not align easily for elimination.

Algebra

Algebra forms the foundation for solving systems of equations. It involves using operations like addition, subtraction, multiplication, and division to manipulate equations and isolate variables.
In solving equations:

• It’s crucial to perform the same operation on both sides of the equation to maintain balance.
• To isolate a variable, undo operations around it, moving terms across the equal sign when necessary.

Algebra simplifies complex mathematical problems and helps in understanding relationships between different quantities or variables.

Coordinate Plane

The coordinate plane is a two-dimensional surface defined by an x-axis and a y-axis, where each point is identified by a pair of numbers. This plane allows us to graphically represent systems of equations and find their solutions visually.
Key concepts include:

• The x-axis and y-axis intersect at the origin (0,0).
• Linear equations can be graphed as straight lines.
• The solution to a system of equations is where their graphs intersect.

Using a coordinate plane helps in visualizing the relationship between equations and interpreting the overlap as the solution to the system.