Consider the equation $2y - 4x = 12$. Which equation, when graphed with the given equation, will form a system with one solution?
A. $-y - 2x = 6$
B. $-y + 2x = 12$
C. $y = 2x + 6$
D. $y = 2x + 12$
Answer (2)
Rewrite the given equation 2 y − 4 x = 12 in slope-intercept form: y = 2 x + 6 .
Analyze each answer choice to find an equation with a different slope.
Option A, − y − 2 x = 6 , can be rewritten as y = − 2 x − 6 , which has a slope of -2.
Since the slopes are different, the system has one solution. The answer is − y − 2 x = 6 .
Explanation
Understanding the Problem We are given the equation 2 y − 4 x = 12 and asked to find another equation that, when graphed with the given equation, will form a system with one solution. This means the two lines must intersect at exactly one point. In other words, the lines must have different slopes.
Rewriting the Given Equation First, 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.
Isolating y 2 y − 4 x = 12 Add 4 x to both sides: 2 y = 4 x + 12 Divide both sides by 2: y = 2 x + 6
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.
Analyzing Option A A) − y − 2 x = 6 Add 2 x to both sides: − y = 2 x + 6 Multiply both sides by -1: y = − 2 x − 6 The slope of this line is -2, which is different from the slope of the given equation (2). Therefore, this system has one solution.
Analyzing Option B B) − y + 2 x = 12 Subtract 2 x from both sides: − y = − 2 x + 12 Multiply both sides by -1: y = 2 x − 12 The slope of this line is 2, which is the same as the slope of the given equation. The y-intercept is -12, which is different from the y-intercept of the given equation (6). Therefore, this system has no solution.
Analyzing Option C C) y = 2 x + 6 The slope of this line is 2, which is the same as the slope of the given equation. The y-intercept is 6, which is the same as the y-intercept of the given equation. Therefore, this system has infinitely many solutions.
Analyzing Option D D) y = 2 x + 12 The slope of this line is 2, which is the same as the slope of the given equation. The y-intercept is 12, which is different from the y-intercept of the given equation (6). Therefore, this system has no solution.
Final Answer The equation that forms a system with one solution is − y − 2 x = 6 , which can be rewritten as y = − 2 x − 6 .
Examples
Understanding systems of equations is crucial in various real-world applications. For instance, consider a scenario where you're trying to determine the break-even point for a business. You might have one equation representing the cost of production and another representing the revenue generated from sales. The point where these two equations intersect (the solution to the system) indicates the level of sales needed to cover all costs. Similarly, in physics, systems of equations can be used to analyze the motion of objects or the flow of electricity in circuits. By understanding how to solve systems of equations, you can model and analyze a wide range of real-world phenomena.
The equation that, when graphed with 2 y − 4 x = 12 , forms a system with one solution is − y − 2 x = 6 . This equation has a different slope (-2) compared to the original equation (slope of 2). Hence, they will intersect at one point.
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