
When it comes to solving systems of linear equations, the goal is often to find the point where two lines intersect. This point, known as the solution, represents the values of the variables that satisfy both equations simultaneously. In this article, we will explore how to sketch a system of two linear equations whose solution is (–1, 3). Along the way, we will delve into the fascinating world of linear equations, their graphical representations, and the infinite possibilities they offer.
Understanding the Basics: What is a Linear Equation?
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form:
[ y = mx + b ]
where:
- ( m ) is the slope of the line,
- ( b ) is the y-intercept, and
- ( x ) and ( y ) are the variables.
When we have two linear equations, we can represent them graphically as two lines on the coordinate plane. The point where these two lines intersect is the solution to the system of equations.
Sketching a System of Two Linear Equations with a Given Solution
To sketch a system of two linear equations whose solution is (–1, 3), we need to ensure that both lines pass through the point (–1, 3). This means that when ( x = -1 ), both equations should yield ( y = 3 ).
Let’s start by choosing two different slopes for our lines. The slope determines the steepness and direction of the line. For simplicity, let’s choose slopes of 2 and –1.
Equation 1: Slope = 2
Using the point-slope form of a linear equation:
[ y - y_1 = m(x - x_1) ]
We can plug in the slope ( m = 2 ) and the point (–1, 3):
[ y - 3 = 2(x - (-1)) ] [ y - 3 = 2(x + 1) ] [ y - 3 = 2x + 2 ] [ y = 2x + 5 ]
So, the first equation is:
[ y = 2x + 5 ]
Equation 2: Slope = –1
Using the same point-slope form with ( m = -1 ) and the point (–1, 3):
[ y - 3 = -1(x - (-1)) ] [ y - 3 = -1(x + 1) ] [ y - 3 = -x - 1 ] [ y = -x + 2 ]
So, the second equation is:
[ y = -x + 2 ]
Graphical Representation
Now that we have our two equations, let’s sketch them on the coordinate plane.
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Equation 1: ( y = 2x + 5 )
- The y-intercept is 5, so the line passes through (0, 5).
- The slope is 2, meaning for every 1 unit increase in ( x ), ( y ) increases by 2 units.
- Plot the point (0, 5) and use the slope to find another point, such as (1, 7).
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Equation 2: ( y = -x + 2 )
- The y-intercept is 2, so the line passes through (0, 2).
- The slope is –1, meaning for every 1 unit increase in ( x ), ( y ) decreases by 1 unit.
- Plot the point (0, 2) and use the slope to find another point, such as (1, 1).
When you plot these two lines, you will see that they intersect at the point (–1, 3), which is the solution to the system.
Exploring Infinite Possibilities
The beauty of linear equations lies in their infinite possibilities. While we have chosen specific slopes for our equations, there are countless other combinations that could also result in the solution (–1, 3). For example, you could choose different slopes or even different forms of linear equations, such as the standard form ( Ax + By = C ).
Moreover, the concept of linear equations extends beyond two dimensions. In higher dimensions, linear equations can represent planes, hyperplanes, and more complex geometric objects. The principles remain the same: finding the intersection points that satisfy all equations in the system.
Applications of Linear Equations
Linear equations are not just abstract mathematical concepts; they have numerous real-world applications. From economics to engineering, linear equations are used to model relationships, predict outcomes, and solve practical problems.
For instance, in economics, linear equations can represent supply and demand curves, where the intersection point determines the equilibrium price and quantity. In engineering, linear equations are used to model systems and predict how they will behave under different conditions.
Conclusion
Sketching a system of two linear equations whose solution is (–1, 3) is a fundamental exercise that helps us understand the relationship between algebraic equations and their graphical representations. By choosing different slopes and forms, we can create an infinite number of systems that all intersect at the same point. This exercise not only reinforces our understanding of linear equations but also opens the door to exploring more complex mathematical concepts and their real-world applications.
Related Q&A
Q1: Can a system of linear equations have more than one solution?
A1: Yes, a system of linear equations can have infinitely many solutions if the equations represent the same line. However, if the lines are parallel and distinct, the system has no solution.
Q2: How do you determine if a system of linear equations has no solution?
A2: If the lines represented by the equations are parallel and do not intersect, the system has no solution. This occurs when the slopes of the lines are equal, but the y-intercepts are different.
Q3: What is the significance of the slope in a linear equation?
A3: The slope of a linear equation determines the steepness and direction of the line. A positive slope indicates that the line rises as ( x ) increases, while a negative slope indicates that the line falls as ( x ) increases.
Q4: Can linear equations be used to model non-linear relationships?
A4: Linear equations are best suited for modeling linear relationships. For non-linear relationships, other types of equations, such as quadratic or exponential, are more appropriate.
Q5: How are linear equations used in machine learning?
A5: In machine learning, linear equations are often used in linear regression models to predict outcomes based on input features. The goal is to find the best-fitting line that minimizes the error between the predicted and actual values.