Graph Of System Of Equations: How To Determine The Solution

Have you ever wondered how to visualize the solution to a system of equations? It's all about understanding the graphs! This article will walk you through how to determine the graphical representation of a system of equations, focusing on identifying whether the lines are parallel, intersecting, or coincident. We'll use the given system as an example and break down the steps to find the correct answer.

Understanding Systems of Equations and Their Graphs

Before diving into the specifics, let's establish a solid foundation. A system of equations is simply a set of two or more equations containing the same variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. Graphically, each equation in a system represents a line. The relationship between these lines tells us about the solutions:

• Intersecting Lines: If the lines intersect at a single point, the system has one unique solution. The coordinates of the intersection point represent the values of the variables that satisfy both equations.
• Parallel Lines: If the lines are parallel, they never intersect. This means there is no solution that satisfies both equations, and the system has no solution.
• Coincident Lines: If the lines are coincident (they overlap completely), every point on the line represents a solution. In this case, the system has infinitely many solutions.

So, when faced with the question of how to describe the graph of a given system of equations, the core task is to figure out the spatial relationship between the lines represented by those equations. Are they crossing paths, running alongside without ever meeting, or are they essentially the same line in disguise?

Analyzing the Given System of Equations

Let's consider the system of equations presented:

$egin{array}{l}

1. 5 x+0.2 y=2.68 \
2. 6 x+0.3 y=2.08 \end{array}$

To determine the relationship between these lines, we can use a couple of approaches. One method is to rewrite each equation in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. Comparing the slopes and y-intercepts will tell us if the lines are parallel, intersecting, or coincident. Another method involves solving the system algebraically to see if we obtain a unique solution, no solution, or infinitely many solutions.

Method 1: Slope-Intercept Form

Let's rewrite the equations in slope-intercept form:

Equation 1:

$1. 5x + 0.2y = 2.68$

Subtract 1.5x from both sides:

$2. 2y = -1.5x + 2.68$

Divide both sides by 0.2:

$y = -7.5x + 13.4$

Equation 2:

$3. 6x + 0.3y = 2.08$

Subtract 1.6x from both sides:

$4. 3y = -1.6x + 2.08$

Divide both sides by 0.3:

y = -rac{16}{3}x + rac{208}{30} ewline y eq -7.5x + 6.93

Now we can clearly see the slopes and y-intercepts:

• Equation 1: Slope = -7.5, y-intercept = 13.4
• Equation 2: Slope = -16/3 ≈ -5.33, y-intercept = 208/30 ≈ 6.93

Since the slopes are different (-7.5 ≠ -5.33), the lines are not parallel and not coincident. They must intersect at a single point.

Method 2: Solving the System Algebraically

We can use either substitution or elimination to solve the system. Let's use elimination. To eliminate y, we can multiply the first equation by -1.5 and the second equation by 1:

• Equation 1 (multiplied by -1.5): $-2.25x - 0.3y = -4.02$
• Equation 2 (multiplied by 1): $1.6x + 0.3y = 2.08$

Now, add the two equations together:

$(-2.25x - 0.3y) + (1.6x + 0.3y) = -4.02 + 2.08$

$-0.65x = -1.94$

Divide both sides by -0.65:

$x = 2.98$

Now, substitute the value of x back into either of the original equations to solve for y. Let's use the first equation:

$5. 5(2.98) + 0.2y = 2.68$

$6. 47 + 0.2y = 2.68$

Subtract 4.47 from both sides:

$7. 2y = -1.79$

Divide both sides by 0.2:

$y = -8.95$

Since we obtained a unique solution (x ≈ 2.98, y ≈ -8.95), the lines intersect at a single point. This confirms our conclusion from the slope-intercept method.

Determining the Intersection Point

While we know the lines intersect, let's examine the provided options to see if we can pinpoint the exact intersection point.

The options given are:

A. The lines are parallel. B. The lines intersect at (1.6, 1.4). C. The lines intersect at (3.1, 0.5). D.

We've already established that the lines are not parallel, so option A is incorrect. Let's check if either of the given intersection points (1.6, 1.4) or (3.1, 0.5) satisfy both equations.

Checking (1.6, 1.4):

• Equation 1: 1.5(1.6) + 0.2(1.4) = 2.4 + 0.28 = 2.68 (Correct)
• Equation 2: 1.6(1.6) + 0.3(1.4) = 2.56 + 0.42 = 2.98 ≠ 2.08 (Incorrect)

Therefore, (1.6, 1.4) is not the intersection point.

Checking (3.1, 0.5):

• Equation 1: 1.5(3.1) + 0.2(0.5) = 4.65 + 0.1 = 4.75 ≠ 2.68 (Incorrect)
• Equation 2: 1.6(3.1) + 0.3(0.5) = 4.96 + 0.15 = 5.11 ≠ 2.08 (Incorrect)

Therefore, (3.1, 0.5) is also not the intersection point.

It seems there might be a slight error in the provided options or the question itself, as our calculations using both slope-intercept form and algebraic methods indicate that the lines intersect, but neither of the provided intersection points is correct. Our algebraic solution yielded x ≈ 2.98 and y ≈ -8.95, which is significantly different from the provided options.

Conclusion

In summary, by analyzing the slopes and y-intercepts, or by solving the system of equations algebraically, we determined that the graph of the given system of equations consists of two lines that intersect at a single point. However, neither of the provided intersection points (1.6, 1.4) or (3.1, 0.5) is the correct solution. Based on our calculations, the actual intersection point is approximately (2.98, -8.95).

Understanding how to interpret the graphs of systems of equations is a fundamental skill in algebra. It allows you to visualize the solutions and understand the relationships between different equations. Whether you're dealing with linear systems or more complex equations, the graphical approach provides a powerful tool for problem-solving.

For further exploration of systems of equations and their graphical representations, you might find helpful resources on websites like Khan Academy's Systems of Equations Section.  This resource offers comprehensive explanations, examples, and practice problems to solidify your understanding.