Let's go through each question one by one.
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Description of the Graph of the Linear Inequality \( y \geq 7x - 4 \):
- The inequality \( y \geq 7x - 4 \) has a solid line because the inequality is "greater than or equal to" (≥).
- The y-intercept is -4, and the slope is 7 (which means for every 1 unit increase in x, y increases by 7).
- Since it is greater than or equal to, we shade above the line.
The correct choice is: The graph will be a solid line with a y-intercept of negative four and a slope of seven. The graph will be shaded above the line.
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Value of y in the Solution to the System of Equations:
The original equations are: \[ \frac{1}{3}x + \frac{1}{4}y = 1 \] \[ 2x - 3y = -30 \]
To find the value of y, we can solve the system using substitution or elimination. However, after solving the equations or graphing each, the value of y can be determined.
Solving this system, we get:
- Substitute \(x = -12\) (found from eliminating y).
- Find \(y\) using either equation.
- Eventually, we find that \(y = -3\).
After checking all the supplied answers, the value of \(y\) is: -3 (correct value).
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Value of m for Parallel Lines: We need to set the slopes equal to make the lines parallel. Given the line equation \(8x - 4y = 12\), we rewrite it in slope-intercept form (y = mx + b):
Rewriting gives: \[ -4y = -8x + 12 \implies y = 2x - 3 \]
The slope here is 2.
Thus, for the second line \(y = mx\) to be parallel, we require: \[ m = 2 \]
So the correct value of m for parallel lines is: 2.
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Approximate Solution to the System of Equations: The equations given are: \[ y = 0.5x + 3.5 \] \[ y = -\frac{2}{3}x + \frac{1}{3} \] By graphing these two lines or solving them, we find the intersection point.
Upon inspecting the given points and looking at where they intersect, we find the approximate solution: (–2.7, 2.1).
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Graph Representing the Wholesaler's Ordering Condition: Since the wholesaler requires a minimum of 4 items in total, and the items are represented by \(x\) (sofas) and \(y\) (pairs of pillows), the equation representing that scenario would be:
\[ x + y \geq 4 \]
The correct graph would be: A solid line that has a positive slope and shade above the line (total must be at least 4).
Among the given descriptions: It should be a solid line with section above it marked. Hence: Graph possibly is "3" representing \(x + y = 4\).
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System of Linear Inequalities Represented by the Graph: We need to analyze the first and second lines. The first line is \(y = x - 2\) (positive slope, shaded to the left). The second line appears as if it is something similar to \(x + 2y < 4\) which would shade downward and to the left, suggesting another negative slope.
Thus, the system could be: y > x - 2 and x + 2y < 4.
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Solution to the System of Equations: Given:
\[ y = -5x + 3 \] \[ y = 1 \]
Setting them equal gives: \[ -5x + 3 = 1 \implies 5x = 2 \implies x = \frac{2}{5} \]
and \[ y = 1 \]
Checking points given with (0.4, 1), solving confirms the solution as (0.4, 1).
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Steps to Eliminate y-terms: To eliminate \(y\), we need to make the coefficients of \(y\) equal (but opposite).
First equation: \(4x - 3y = 34\) ⇒ is \(3\). Second equation: \(3x + 2y = 17\) ⇒ is \(2\).
Therefore, each equation multiplied gives: The first equation should be multiplied by 2 and the second by 3.
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Ordered Pair in the Solution Set of Inequalities: As per the inequalities given: \[ y > \frac{3}{2}x - 1 \] \[ y < \frac{3}{2}x - 1 \]
From the line formed, it is clear \(y\) must satisfy both. From the options: Only ordered pairs above or below the lines can be in solution.
Checking each option, such as: No ordered pair in the given values fits. Hence: No solution.
- Inequality to Represent Purchases: Gabriela bought more than 12 items. Hence, the equation would be represented as:
\( x + y > 12 \)
So the inequality that best represents her purchase is: x + y > 12.
If you need any further assistance or clarifications, let me know!
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