solve this---**Textual Information:** 1. Given $y = 3x - 5$, which of the following is true? View answer choices

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Let's analyze the linear equation y equals 3x minus 5. This equation is in slope-intercept form, y equals mx plus b, where m is the slope and b is the y-intercept. From our equation, we can identify that the slope m equals 3, and the y-intercept b equals negative 5. This means the line crosses the y-axis at the point (0, negative 5). We can verify that the point (1, negative 2) lies on this line by substituting x equals 1 into our equation: y equals 3 times 1 minus 5, which equals negative 2. The slope of 3 means that for every 1 unit increase in x, y increases by 3 units, as shown by the slope triangle on the graph. Now let's test whether specific points lie on our line y equals 3x minus 5. For the point (2, 1), we substitute x equals 2 into our equation: y equals 3 times 2 minus 5, which equals 6 minus 5, which equals 1. Since the y-coordinate matches our calculation, the point (2, 1) is on the line. Similarly, for the point (3, 4), we get y equals 3 times 3 minus 5, which equals 9 minus 5, which equals 4. This point is also on the line. However, for the point (0, 0), we get y equals 3 times 0 minus 5, which equals negative 5. Since this doesn't match the y-coordinate of 0, the point (0, 0) is not on the line. We can see on the graph that points (2, 1) and (3, 4) lie exactly on the line, while (0, 0) is 5 units above the actual y-intercept. Let's examine the key properties of our linear equation y equals 3x minus 5. First, the slope is positive 3, meaning the line rises as x increases - for every 1 unit increase in x, y increases by 3 units. Second, the y-intercept is negative 5, so the line crosses the y-axis at the point (0, negative 5). Third, to find the x-intercept, we set y equal to 0 and solve: 0 equals 3x minus 5, which gives us x equals 5/3 or approximately 1.67. Fourth, any line parallel to our original line must have the same slope of 3, such as y equals 3x plus 2. Finally, perpendicular lines have slopes that are negative reciprocals of each other, so a line perpendicular to our original line would have a slope of negative 1/3, like y equals negative x/3 plus 1. Notice how the perpendicular line crosses our original line at a right angle. Linear equations like y equals 3x minus 5 have many real-world applications. First, in cost analysis, the equation C of x equals 5 plus 3x represents a total cost function with a fixed cost of $5 and a variable cost of $3 per unit. The y-intercept of 5 represents the fixed cost you pay regardless of quantity, while the slope of 3 represents the per-unit cost. Second, in distance-time relationships, the equation d of t equals negative 5 plus 3t could represent position, where you start 5 miles behind a reference point and move at 3 miles per hour. The negative y-intercept represents your initial position, and the positive slope represents your speed. Third, temperature conversion follows a similar pattern - the Fahrenheit to Celsius conversion formula F equals nine-fifths C plus 32 is also a linear equation. Notice how the cost and distance functions intersect at the point (5, 20), which could represent a break-even point or some other significant threshold in a real application. Let's summarize what we've learned about the linear equation y equals 3x minus 5. First, this equation is written in slope-intercept form, which is y equals mx plus b, making it easy to identify its key properties. Second, the slope m equals 3, which tells us that y increases by 3 units for every 1 unit increase in x, giving the line its positive upward slant. Third, the y-intercept b equals negative 5, meaning the line crosses the y-axis at the point (0, negative 5). Fourth, we calculated the x-intercept to be x equals 5/3 or approximately 1.67, which is where the line crosses the x-axis. Finally, we can determine whether any point lies on this line by substituting its x-coordinate into our equation and checking if the resulting y-value matches the point's y-coordinate. These properties help us understand the behavior of linear equations and apply them to real-world situations like cost analysis, distance-time relationships, and temperature conversions.