Let's address each question based on the provided equations and information.
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Finding the solution to the system of equations: The given equations are:
- \( y = 4x - 10 \)
- \( y = -2x + 8 \)
To find the solution, set the equations equal to each other: \[ 4x - 10 = -2x + 8 \]
Now, solve for \( x \): \[ 4x + 2x = 8 + 10 \] \[ 6x = 18 \] \[ x = 3 \]
Now substitute \( x = 3 \) back into either equation to find \( y \): \[ y = 4(3) - 10 = 12 - 10 = 2 \]
Therefore, the solution to the system is \( (3, 2) \).
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Estimate the solution based on the graph of:
- \( y = 2x + 3 \)
- \( y = 12x + 6 \)
Based on a graphical estimation (which we can't see, but making an educated guess), we can evaluate the possible responses. The point that seems closest to both lines' point of intersection is:
- (2, 7) — likely represents the correct estimate based on the typical slopes.
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Estimate the solution based on the graph of:
- \( y = -x + 6 \)
- \( y = 3x - 2 \)
Again, without the actual graph visible, one would estimate the intersection point. Typically, one can visually determine or calculate as needed.
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Determine the number of solutions for: \[ 2x + y = 6 \] \[ 4x + 2y = 8 \]
By manipulating the second equation: Divide the entire equation \( 4x + 2y = 8 \) by 2: \[ 2x + y = 4 \]
Now compare it to the first equation: \[ 2x + y = 6 \quad \text{(first equation)} \] \[ 2x + y = 4 \quad \text{(transformed second equation)} \]
Since both equations represent different constant values (6 vs 4) for the same linear relationship, these lines are parallel and do not intersect. Thus, there are 0 solutions.
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When a system of equations involves two lines that __________, there will be infinite solutions. The correct term to fill in the blank is are identical. If two lines represent the same relationship (meaning they have the same slope and y-intercept), they will overlap entirely, leading to infinitely many solutions.
Summary Answers:
- Solution of (3, 2)
- Estimate (2, 7)
- Further graph estimation needed.
- 0 solutions
- are identical