A system of equations has 1 solution. If [tex]4 x-y=5[/tex] is one of the equations, which could be the other equation?
[tex]y=-4 x+5[/tex]
[tex]y=4 x-5[/tex]
[tex]2 y=8 x-10[/tex]
[tex]-2 y=-8 x-10[/tex]
Answer (2)
Rewrite the given equation 4 x − y = 5 in slope-intercept form: y = 4 x − 5 .
Rewrite each answer choice in slope-intercept form and compare the slopes.
A system has one solution if the slopes are different.
The equation y = − 4 x + 5 has a different slope, so the answer is y = − 4 x + 5 .
Explanation
Understanding the Problem We are given one equation of a system of equations: 4 x − y = 5 . We need to determine which of the other given equations would result in the system having exactly one solution. A system of two linear equations has one solution if the lines are not parallel, meaning they have different slopes.
Finding the Slope and Intercept of the Given Equation Let's rewrite the given equation in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept. Starting with 4 x − y = 5 , we can rearrange it to isolate y :
4 x − y = 5
− y = − 4 x + 5
y = 4 x − 5
So, the given equation has a slope of 4 and a y-intercept of − 5 .
Analyzing the Answer Choices Now, let's examine each of the answer choices and rewrite them in slope-intercept form to determine their slopes and y-intercepts:
y = − 4 x + 5 : This equation has a slope of − 4 and a y-intercept of 5 . Since the slope is different from the given equation's slope ( 4 ), this system would have one solution.
y = 4 x − 5 : This equation has a slope of 4 and a y-intercept of − 5 . Since both the slope and y-intercept are the same as the given equation, this represents the same line, and the system would have infinitely many solutions.
2 y = 8 x − 10 : Divide both sides by 2 to get y = 4 x − 5 . This equation has a slope of 4 and a y-intercept of − 5 . Since both the slope and y-intercept are the same as the given equation, this represents the same line, and the system would have infinitely many solutions.
− 2 y = − 8 x − 10 : Divide both sides by − 2 to get y = 4 x + 5 . This equation has a slope of 4 and a y-intercept of 5 . Since the slope is the same as the given equation, but the y-intercept is different, the system would have no solution (parallel lines).
Determining the Correct Equation The only equation that results in a system with one solution is y = − 4 x + 5 , because it has a different slope ( − 4 ) than the given equation ( 4 ).
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if you have a cost equation y = 2 x + 100 (where x is the number of units produced and y is the total cost) and a revenue equation y = 5 x , solving this system of equations will tell you how many units you need to sell to cover your costs. The solution represents the point where the cost and revenue lines intersect, indicating the break-even point. Understanding how to solve systems of equations is crucial for making informed business decisions.
The equation that allows the system to have exactly one solution is y = − 4 x + 5 because it has a different slope than the given equation 4 x − y = 5 . The original equation has a slope of 4 , while this option has a slope of − 4 , making them non-parallel. Hence, they intersect at one point, leading to a unique solution.
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