Graph Two Lines Whose Solution Is 1 4

Graphing Two Lines Whose Solution is (1, 4)

This article delves into the fascinating world of linear equations and their graphical representations, specifically focusing on how to create two lines that intersect at the point (1, 4). We'll explore various methods, discuss the underlying concepts, and offer practical examples to solidify your understanding. This comprehensive guide will be beneficial for students learning algebra, as well as anyone seeking a refresher on graphing linear equations and solving systems of equations.

Understanding Linear Equations and Their Graphs

Before we dive into creating lines that intersect at (1, 4), let's review the fundamentals of linear equations and their graphical representations.

A linear equation is an equation that, when graphed, produces a straight line. It's typically expressed in the form:

y = mx + b

Where:

• y represents the dependent variable (the output).
• x represents the independent variable (the input).
• m represents the slope of the line (the rate of change of y with respect to x).
• b represents the y-intercept (the point where the line crosses the y-axis).

The slope (m) indicates the steepness and direction of the line. A positive slope indicates an upward-sloping line, while a negative slope indicates a downward-sloping line. A slope of zero results in a horizontal line.

The y-intercept (b) is the value of y when x is 0. It's the point where the line intersects the y-axis.

Graphing a linear equation involves plotting points that satisfy the equation on a coordinate plane and then drawing a line through those points. We can find points by substituting different values of x into the equation and solving for y.

Creating Two Lines Intersecting at (1, 4)

Our goal is to find two distinct linear equations whose graphs intersect at the point (1, 4). This means that the coordinates (x = 1, y = 4) must satisfy both equations. There are infinitely many pairs of lines that meet this condition. Let's explore several methods to generate these equations.

Method 1: Using the Point-Slope Form

The point-slope form of a linear equation is:

y - y₁ = m(x - x₁)

Where:

• (x₁, y₁) is a point on the line.
• m is the slope of the line.

Since we know the point (1, 4) lies on both lines, we can use this form to construct our equations. We just need to choose different slopes for each line.

Let's choose a slope of 2 for the first line:

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

Now, let's choose a slope of -1 for the second line:

y - 4 = -1(x - 1) y - 4 = -x + 1 y = -x + 5

These two lines, y = 2x + 2 and y = -x + 5, intersect at the point (1, 4). You can verify this by substituting x = 1 into each equation and confirming that y = 4 in both cases.

Method 2: Using the Slope-Intercept Form and Different Slopes

We can also use the slope-intercept form (y = mx + b) directly. Since the point (1, 4) must satisfy both equations, we can substitute x = 1 and y = 4 to find the y-intercept (b) for each line, given a chosen slope (m).

Let's choose a slope of 3 for the first line:

4 = 3(1) + b b = 1

So our first equation is: y = 3x + 1

Let's choose a slope of -2 for the second line:

4 = -2(1) + b b = 6

So our second equation is: y = -2x + 6

Again, both lines y = 3x + 1 and y = -2x + 6 intersect at (1, 4).

Method 3: System of Equations and Manipulation

We can start with a system of equations and manipulate them to ensure they intersect at (1, 4). Let's begin with a simple example:

x + y = 5 x - y = -3

Solving this system using either substitution or elimination will yield the solution (1, 4). You can verify this by substituting x = 1 and y = 4 into both equations. This method showcases how a system of equations can be designed to have a specific solution.

Method 4: Using Parallel Lines and a Transversal

While the above methods create intersecting lines, we can also explore a slightly more complex scenario. Consider two parallel lines and a transversal line that intersects them. The point of intersection between the transversal and one of the parallel lines can be (1, 4).

For example:

Let's say our parallel lines are: y = 4 (horizontal line) x = 1 (vertical line)

The transversal can be any line that passes through (1, 4) but isn't parallel to either of the lines. For example, we could use y = 2x + 2 again. In this case, the point (1,4) is where the transversal intersects the horizontal line y = 4. Although not a typical intersection of two lines in the traditional sense, this demonstrates the concept of intersection within a system of lines.

Graphical Representation and Verification

The best way to visualize these solutions is by graphing the lines. You can use graphing paper, a graphing calculator, or online graphing tools to plot each pair of equations we derived. In each case, you should observe that the lines intersect precisely at the point (1, 4). This visual confirmation reinforces the mathematical calculations.

Significance and Applications

Understanding how to create lines intersecting at a specific point has numerous applications in various fields:

• Computer Graphics: Precise line intersections are crucial in computer graphics for tasks such as creating 3D models and rendering images.

• Engineering and Physics: Solving systems of linear equations, represented graphically by line intersections, is essential in engineering and physics for analyzing physical systems and modeling behavior.

• Economics and Finance: Linear equations and their graphical representations are commonly used to model economic relationships and predict trends. The point of intersection might represent an equilibrium point or a critical juncture.

• Data Analysis: In data analysis, line intersections can indicate significant correlations or points of convergence in data sets.

Further Exploration and Challenges

This article provides a foundational understanding of generating lines that intersect at a specified point. For further exploration, consider these challenges:

• Find three lines intersecting at (1, 4). Extend the methods discussed to create a system involving three lines.

• Explore non-linear equations. Consider how the concept of intersection extends to curves and other non-linear functions.

• Investigate systems of inequalities. Explore how the solution point relates to the overlapping regions defined by inequalities.

• Use different coordinate systems. Consider how the same principles apply when using polar coordinates or other coordinate systems.

By understanding the fundamental principles of linear equations and their graphical representation, you can effectively create lines intersecting at any desired point, opening up a world of possibilities in mathematics and its diverse applications. Remember, practice is key to mastering these concepts. Experiment with different slopes and y-intercepts to create your own systems of equations and visualize their intersections.