Solving Systems Of Equations Finding The Unique Solution

When we delve into the world of systems of equations, we often encounter scenarios where two or more equations are intertwined, sharing variables and seeking common solutions. A system of equations is said to have one solution when there is a unique set of values for the variables that satisfies all equations simultaneously. Geometrically, in a two-variable system, this corresponds to the point where two lines intersect on a graph. Understanding the conditions that lead to a single solution is crucial in various fields, from mathematics and physics to economics and engineering.

To grasp the concept fully, let's consider a system of two linear equations in two variables, such as 'x' and 'y'. The general form of a linear equation is Ax + By = C, where A, B, and C are constants. When we have two such equations, the system can exhibit three possible scenarios: one solution, no solution, or infinitely many solutions. The key to determining the number of solutions lies in the relationship between the coefficients of the variables and the constants in the equations.

When a system has one unique solution, the lines represented by the equations intersect at a single point. This implies that the slopes of the two lines are different. The slope of a line in the form Ax + By = C can be found by rearranging the equation into slope-intercept form, which is y = mx + b, where 'm' represents the slope. If the slopes of the two lines are not equal, the lines will intersect, and the system will have one solution. Conversely, if the slopes are equal but the y-intercepts ('b' values) are different, the lines are parallel and will never intersect, resulting in no solution. If both the slopes and the y-intercepts are the same, the lines coincide, and there are infinitely many solutions.

Consider the given equation, 4x - y = 5. Our task is to find another equation that, when paired with this one, will result in a system with exactly one solution. This means we need an equation that represents a line with a different slope than the line represented by 4x - y = 5. To determine the slope of the given line, we can rearrange it into slope-intercept form:

4x - y = 5 -y = -4x + 5 y = 4x - 5

From this, we can see that the slope of the given line is 4. Therefore, any equation that has a slope different from 4, when expressed in slope-intercept form, will create a system with one solution when paired with the given equation. Let's examine the provided options to identify the one that fits this criterion. By analyzing the slopes and intercepts, we can accurately determine the equation that yields a unique solution for the system. Understanding these principles allows us to solve a wide range of problems involving systems of equations, making it a fundamental concept in mathematics and its applications.

To identify the equation that, when paired with 4x - y = 5, results in a system with one solution, we need to examine each option and determine its slope. The given equation, 4x - y = 5, has a slope of 4, as we derived in the previous section by converting it to slope-intercept form (y = 4x - 5). Therefore, any equation with a slope different from 4 will create a system with a unique solution.

Let's analyze each option:

• Option A: y = -4x + 5

  This equation is already in slope-intercept form, and we can directly see that its slope is -4. Since -4 is different from the slope of the given equation (which is 4), this equation will intersect the line represented by 4x - y = 5 at a single point. Therefore, a system consisting of these two equations will have one solution. This option appears to be a strong candidate.

• Option B: y = 4x - 5

  This equation also is in slope-intercept form, and its slope is 4. Notice that this is the same slope as the given equation, 4x - y = 5. However, the y-intercepts are also the same (-5 in both equations). This means the two equations represent the same line. If we were to graph these two equations, they would overlap perfectly, indicating infinitely many solutions, not one unique solution. Thus, this option is not the correct answer.

• Option C: 2y = 8x - 10

  To determine the slope of this equation, we need to convert it to slope-intercept form. We can do this by dividing both sides of the equation by 2:

  2y / 2 = (8x - 10) / 2 y = 4x - 5

  After converting, we see that this equation has a slope of 4 and a y-intercept of -5. This is identical to the given equation, 4x - y = 5. As with Option B, these two equations represent the same line and will have infinitely many solutions, not a single solution. Hence, Option C is not the correct answer.

• Option D: -2y = -8x - 10

  Similar to Option C, we need to convert this equation to slope-intercept form to identify its slope. To do this, we divide both sides of the equation by -2:

  -2y / -2 = (-8x - 10) / -2 y = 4x + 5

  This equation has a slope of 4, the same as the given equation. However, the y-intercept is 5, which is different from the y-intercept of the given equation (-5). This means the two lines are parallel, as they have the same slope but different y-intercepts. Parallel lines never intersect, so a system consisting of these two equations will have no solution, not one unique solution. Thus, Option D is not the correct answer.

By analyzing the slopes of each option, we can confidently conclude that Option A is the equation that, when paired with 4x - y = 5, will result in a system with one solution. The different slopes of the lines ensure that they intersect at a single point, fulfilling the condition for a unique solution. This systematic approach to analyzing options is vital in solving mathematical problems involving systems of equations.

In the context of linear equations and systems of equations, the concepts of slope and y-intercept are fundamental. They provide a clear and concise way to understand the behavior and relationship between lines. As we've seen in the previous sections, the slope determines the steepness and direction of a line, while the y-intercept indicates the point where the line crosses the y-axis. Together, these two parameters uniquely define a line in a two-dimensional plane. Understanding their roles is crucial for solving problems related to systems of equations, graphing, and linear relationships in general.

The slope of a line is a measure of how much the line rises (or falls) for every unit increase in the horizontal direction. It is often described as