Question

1. which graph represents the equation 12 = 3x + 4y? explain how you know.
2. clare knows that priya has a bunch of nickels and dimes in her pocket and that the total amount is $1.25.

a. find one possibility for the number of nickels and number of dimes that could be in priyas pocket.
b. write an equation that describes the relationship between the number of dimes and the number of nickels in priyas pocket.
c. explain what the point (13,6) means in this situation.
d. is the point (13,6) a solution to the equation you wrote? explain your reasoning

Explanation:

Step1: Rewrite the equation 12 = 3x + 4y in slope - intercept form

Solve for y:

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The slope \(m =-\frac{3}{4}\) and the y - intercept \(b = 3\). A line with a negative slope will go down from left to right and cross the y - axis at 3. So the first graph is the correct one for the equation 12 = 3x + 4y.

Step2: Solve problem 5a

Let \(n\) be the number of nickels and \(d\) be the number of dimes. We know that a nickel is worth 5 cents or \(\$0.05\) and a dime is worth 10 cents or \(\$0.10\), and \(0.05n+0.10d = 1.25\). Multiply through by 100 to get \(5n + 10d=125\), or \(n + 2d=25\). Let \(d = 10\), then \(n=25 - 2\times10=5\). So one possibility is 5 nickels and 10 dimes.

Step3: Solve problem 5b

The equation based on the value of the coins is \(0.05n+0.10d = 1.25\), or in whole - number form \(5n + 10d=125\) (divide by 5) gives \(n + 2d=25\).

Step4: Solve problem 5c

In the context of the number of nickels and dimes, if we assume \(n\) is on the x - axis and \(d\) is on the y - axis, the point \((13,6)\) means there are 13 nickels and 6 dimes.

Step5: Solve problem 5d

Substitute \(n = 13\) and \(d = 6\) into the equation \(n + 2d=25\).

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Since the left - hand side equals the right - hand side, the point \((13,6)\) is a solution to the equation \(n + 2d=25\).

Answer:

1. The first graph represents the equation 12 = 3x + 4y because when rewritten in slope - intercept form \(y=-\frac{3}{4}x + 3\), it has a negative slope and y - intercept of 3.

5a. One possibility is 5 nickels and 10 dimes.
5b. The equation is \(n + 2d=25\) (where \(n\) is the number of nickels and \(d\) is the number of dimes).
5c. The point \((13,6)\) means there are 13 nickels and 6 dimes.
5d. Yes, the point \((13,6)\) is a solution to the equation \(n + 2d=25\) because \(13+2\times6=25\).