Use only the slopes and y-intercepts of the graphs of the equations to determine the number of solutions of the system: y = 7x + 13 and -21x + 3y = 39. The system has _____. Explai... Use only the slopes and y-intercepts of the graphs of the equations to determine the number of solutions of the system: y = 7x + 13 and -21x + 3y = 39. The system has _____. Explain your reasoning.
Understand the Problem
The question is asking to determine the number of solutions of the system of equations using only the slopes and y-intercepts of the graphs. This involves analyzing the given equations to find their slopes and y-intercepts, then deducing how many points of intersection (solutions) exist based on that information.
Answer
The system has infinitely many solutions.
Answer for screen readers
The system has infinitely many solutions.
Steps to Solve
- Identify slopes and intercepts of the first equation
The first equation is given as:
$$ y = 7x + 13 $$
From this equation, we see:
- Slope (m) = 7
- Y-intercept (b) = 13
- Rearranging the second equation to slope-intercept form
The second equation is:
$$ -21x + 3y = 39 $$
To find the slope and y-intercept, we can solve for ( y ):
$$ 3y = 21x + 39 $$
Dividing all terms by 3, we get:
$$ y = 7x + 13 $$
- Identify slopes and intercepts of the second equation
From the rearranged second equation:
$$ y = 7x + 13 $$
We deduce:
- Slope (m) = 7
- Y-intercept (b) = 13
- Analyze the slopes and intercepts
Both equations share the same slope ( (m = 7) ) and the same y-intercept ( (b = 13) ). This means the lines are identical.
- Determine the number of solutions
Since the two equations represent the same line, they have infinitely many solutions. The graphs intersect at all points along the line.
The system has infinitely many solutions.
More Information
When two linear equations represent the same line, they will coincide at every point along that line. Thus, they have infinitely many intersection points (solutions).
Tips
- Not rearranging equations correctly: Ensure all equations are in slope-intercept form for comparison.
- Misinterpreting the slopes: Remember that equal slopes with different intercepts indicate parallel lines, which have no solutions.
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