Problem 126 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 5 x+2 y=2 \\ -3 x-y=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \(x = -2\) and \(y = 6\), or \( (-2, 6) \).

Step 1: Solve one equation for one variable

Take the second equation \(-3x - y = 0\) and solve for \(y\).\ Add \(3x\) to both sides: \(-y = 3x\).\ Multiply both sides by \(-1\): \(y = -3x\).

Step 2: Substitute into the other equation

Substitute \(y = -3x\) into the first equation \(5x + 2y = 2\).\ The equation becomes \5x + 2(-3x) = 2\.\ Simplify this to get \5x - 6x = 2\.

Step 3: Solve for \(x\)

Combine like terms: \(-x = 2\).\ Multiply both sides by \(-1\) to solve for \(x\): \(x = -2\).

Step 4: Substitute \(x\) back to find \(y\)

Substitute \(x = -2\) back into \(y = -3x\).\ This gives \(y = -3(-2) = 6\).

Step 5: Write the solution as a coordinate

The solution to the system of equations is \(x = -2\) and \(y = 6\). Express this as the coordinate \( (-2, 6) \).

Key Concepts

Substitution Method

The substitution method is a way to solve systems of equations where you solve one equation for one variable, and then substitute that expression into the other equation. This method is particularly useful when one of the equations can be easily solved for one of the variables.
Step 1: First, solve one of the equations for one of the variables. For instance, take the equation \(-3x - y = 0\). Solving for \y\, we get \(y = -3x\).
Step 2: Substitute this result into the other equation. In our example, we substitute \y = -3x\ into \(5x + 2y = 2\), transforming the equation to \(5x + 2(-3x) = 2\).
Step 3: Solve the new equation for the remaining variable. For \(5x - 6x = 2\), combine like terms to get \(-x = 2\), which simplifies to \(x = -2\).
Step 4: Substitute back to find the other variable. Using \x = -2\ in \(y = -3x\), we find \(y = 6\).
Step 5: Write the solution as an ordered pair. The solution is \(-2, 6\), indicating the intersection point of the lines on a graph.

Linear Equations

Linear equations are equations where the highest power of the variable is 1. They represent straight lines when graphed on a coordinate plane. In our exercise, both equations \(5x + 2y = 2\) and \(-3x - y = 0\) are linear.
Simple properties of linear equations include:

• They have one or more variables.
• Their graph is always a straight line.
• They can be written in different forms such as the standard form (Ax + By = C) and slope-intercept form (y = mx + b).

To solve a system of linear equations, we find the point where the two lines intersect. This can be done using various methods like substitution, elimination, or graphing.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is the origin, denoted as (0, 0).
In the context of solving systems of equations, the coordinate plane is crucial for visualizing the solutions. Each point on the plane represents an ordered pair (x, y).
Key points about the coordinate plane:

• The x-coordinate indicates the position along the horizontal axis.
• The y-coordinate indicates the position along the vertical axis.
• A system of equations solution, like our \(-2, 6\), shows where two lines intersect on this plane.

Understanding the coordinate plane helps in graphing equations and identifying their intersection, which represents the solution to the system.