Create a second equation so that the following system has no solution: y = (3/4)x - 4, ?
Understand the Problem
The question provides one linear equation and asks us to create a second linear equation such that the system of equations has no solution. This means the two lines must be parallel but have different y-intercepts. Parallel lines have the same slope.
Answer
$y = 3x + 2$
Answer for screen readers
$y = 3x + 2$
Steps to Solve
- Identify the slope of the given equation
The given equation is $y = 3x + 5$. This is in slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Therefore, the slope of the given equation is $3$.
- Create a new equation with the same slope
To ensure the new equation represents a parallel line, it must have the same slope as the given equation. So, the new equation will have the form $y = 3x + c$, where $c$ is a constant.
- Choose a different y-intercept
To ensure the lines are parallel but not the same line (i.e., to ensure there's no solution), the y-intercept of the new equation must be different from the y-intercept of the original equation. The original equation has a y-intercept of $5$, so we can choose any other value for $c$. Let's choose $c = 2$.
- Write the new equation
The new equation is $y = 3x + 2$. This equation has the same slope as the original equation ($3$), but a different y-intercept ($2$), so the system of equations will have no solution.
$y = 3x + 2$
More Information
Any equation of the form $y = 3x + c$, where $c \ne 5$, will result in a system of equations with no solution. This is because the lines will be parallel and distinct.
Tips
A common mistake is to create an equation with a different slope. If the slopes are different, the lines will intersect, and the system will have one solution. Another mistake is to create an equation with the same slope and the same y-intercept. This creates the same line, and the system has infinitely many solutions.
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