Which values of a and b make this system of equations have infinitely many solutions? 6x + 3y = -15, y = ax + b.
Understand the Problem
The question is asking for specific values of 'a' and 'b' in the equation given that will result in the system having infinitely many solutions. This typically occurs when the two equations are dependent on each other, meaning they represent the same line.
Answer
The values of $a$ and $b$ are $a = -2$ and $b = -5$.
Answer for screen readers
The values of $a$ and $b$ that make the system have infinitely many solutions are $a = -2$ and $b = -5$.
Steps to Solve
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Rearranging the first equation
We start with the first equation:
$$ 6x + 3y = -15 $$
To get it into the slope-intercept form ($y = mx + b$), we isolate $y$: $$ 3y = -6x - 15 $$
Then we divide by 3:
$$ y = -2x - 5 $$ -
Identifying the slope and y-intercept
From the rearranged equation $y = -2x - 5$, we can see that the slope ($m$) is -2 and the y-intercept ($b$) is -5. -
Setting the equations equal
For the second equation $y = ax + b$ to have infinitely many solutions, it must have the same slope and intercept as the first equation. Thus, we set:
- The slope $a = -2$
- The y-intercept $b = -5$
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Conclusion
The values that result in infinitely many solutions are:
$$ a = -2, b = -5 $$
The values of $a$ and $b$ that make the system have infinitely many solutions are $a = -2$ and $b = -5$.
More Information
When two equations yield the same line on a graph, they have infinitely many solutions because every point on that line is a solution. Dependent equations confirm this relationship.
Tips
- Confusing the coefficients: Make sure to match both the slope and y-intercept.
- Forgetting to manipulate both equations into slope-intercept form before comparison.
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