Solving Systems of Linear Equations with y = y

In the world of algebra, solving systems of linear equations is a fundamental skill. These systems represent a set of equations with multiple variables, and our goal is to find the values of those variables that satisfy all equations simultaneously. One common approach to solving systems involves setting the equations equal to each other, particularly when both equations are already expressed in terms of the same variable. This is where the 'y = y' method comes into play.

Understanding the Concept

When two linear equations are both set equal to 'y', it means that their expressions on the right-hand side represent the same value for any given 'x'. This provides a direct way to solve for 'x'.

Example:

Let's consider the following system of equations:

y = 2x + 1
y = -x + 4

Since both equations are equal to 'y', we can set their right-hand sides equal to each other:

2x + 1 = -x + 4

Now, we can solve for 'x':

Combine 'x' terms: 3x + 1 = 4
Subtract 1 from both sides: 3x = 3
Divide both sides by 3: x = 1

We have found that x = 1. To find the corresponding value of 'y', we can substitute this value back into either of the original equations. Let's use the first equation:

y = 2(1) + 1

y = 2 + 1

y = 3

Therefore, the solution to this system of equations is x = 1 and y = 3. This means that the point (1, 3) lies on both lines represented by the equations.

Graphical Interpretation

Graphically, solving a system of linear equations with 'y = y' involves finding the point where the two lines intersect. The 'x' coordinate of the intersection point represents the solution for 'x', and the 'y' coordinate represents the solution for 'y'.

Key Points to Remember:

This method is only applicable when both equations are set equal to 'y'.
After solving for 'x', substitute the value back into either original equation to find 'y'.
The solution represents the point of intersection of the two lines.

Practice Problems:

Try solving these systems of equations using the 'y = y' method:

y = 3x - 2
y = -x + 6

y = -2x + 5
y = 4x - 1

Solving systems of linear equations with 'y = y' is a straightforward and efficient technique. By understanding the concept and practicing with examples, you can master this valuable skill in algebra.

Solving Systems of Linear Equations with y = y