1. Anna has 12 bills in her wallet, some $5 and some $10. The total value of the bills is $100. How many of each bill does Anna have?
2. Tripling the greater of two consecutive even integers gives the same result as subtracting 10 from the lesser even integer. What are the integers?
Fill in the blank with either ALWAYS, SOMETIMES, or NEVER
3. Parallel lines ______________ have the same slope.
4. On a graph, a point on an axis _______________ has a coordinate equal to zero.
5. If the graph of a linear equation Ax + By = C passes through the origin, then C ______________ equals 0.
6. A line with an undefined slope is ____________ the same as a line with at slope of zero.
7. The graph of a function __________ has more than one y-intercept.
8. If the graphs of two linear equations in a system have different slopes, the system __________ has exactly one solution.
9. A system of equations is __________ both inconsistent and dependent.
10. (3 + 17) - 2(3-3) + 1000 =
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1. To solve this problem, we can set up a system of equations. Let x represent the number of $5 bills and y represent the number of $10 bills. We have two equations:
x + y = 12 (equation 1)
5x + 10y = 100 (equation 2)
From equation 1, we can express x in terms of y:
x = 12 - y
Substituting this expression into equation 2, we get:
5(12 - y) + 10y = 100
Expanding and simplifying:
60 - 5y + 10y = 100
5y = 40
y = 8
Substituting this value back into equation 1, we can solve for x:
x + 8 = 12
x = 4
So Anna has 4 $5 bills and 8 $10 bills.
2. Let's represent the consecutive even integers as x and x+2. According to the problem, we have the equation:
3(x+2) = x - 10
Expanding and simplifying:
3x + 6 = x - 10
2x = -16
x = -8
So the consecutive even integers are -8 and -6.
3. ALWAYS. Parallel lines have the same slope. This is a fundamental property of parallel lines.
4. SOMETIMES. A point on the x-axis has a y-coordinate equal to zero and a point on the y-axis has an x-coordinate equal to zero. However, a point on the origin (0,0) has both x and y coordinates equal to zero.
5. ALWAYS. If a graph passes through the origin (0,0), then plugging in x=0 and y=0 into the equation Ax + By = C will give us C = 0. Hence, C always equals 0.
6. NEVER. A line with an undefined slope is a vertical line, and a line with a slope of zero is a horizontal line. These two types of lines are not the same. A vertical line has no defined slope, while a horizontal line has a slope of zero.
7. NEVER. The graph of a function can have at most one y-intercept. By definition, a function maps each value of x to a unique value of y, so there can only be one y-intercept.
8. ALWAYS. If the graphs of two linear equations in a system have different slopes, the lines will intersect at exactly one point, giving us one solution for the system of equations.
9. NEVER. A system of equations cannot be both inconsistent and dependent at the same time. An inconsistent system has no solution, while a dependent system has infinite solutions.
10. The given expression simplifies as follows:
(3 + 17) - 2(3-3) + 1000
Evaluating inside the parentheses:
20 - 2(0) + 1000
Simplifying the multiplication:
20 + 0 + 1000
Combining like terms:
1020
So the answer is 1020.