1.
Find the point of intersection for the systems of equations below.
Give your answer in (x, y) form.
2.
Find the point of intersection for the systems of equations below.
Give your answer in (x, y) form.
Explanation
The given answer (-3,-4),(-3, -4) indicates that the two equations intersect at the point (-3,-4). This means that both equations have the same x-coordinate (-3) and y-coordinate (-4). Therefore, the point of intersection for the systems of equations is (-3,-4).
3.
Find the point of intersection for the systems of equations below.
Give your answer in (x, y) form.
4.
Find the point of intersection for the systems of equations below.
Give your answer in (x, y) form.
Explanation
The given answer is (6, 9). This means that the two systems of equations intersect at the point (6, 9).
5.
Find the point of intersection for the systems of equations below.
Give your answer in (x, y) form.
Explanation
The given answer (-2,5),(-2,5) is the point of intersection for the systems of equations. This means that both equations in the system have the same solution, which is (-2,5).
6.
Find the point of intersection for the systems of equations below.
Give your answer in (x, y) form.
7.
Find the point of intersection for the systems of equations below.
Give your answer in (x, y) form.
Explanation
The phrase "no solution" means that the two systems of equations do not intersect or have any common point. This can occur when the lines represented by the equations are parallel or when they are the same line. In either case, there is no point of intersection, hence the answer is "no solution".
8.
Find the point of intersection for the systems of equations below.
Give your answer in (x, y) form.
9.
Janelle has $20 and is saving $6 per week. April has $150 and is spending $4 per week. When will they both have the same amount of money?
For your answer, fill in the blank below.
They will have the same amount of money in ______.
Explanation
Janelle is saving $6 per week, while April is spending $4 per week. This means that Janelle's savings are increasing by $6 each week, while April's money is decreasing by $4 each week. The difference between their savings is $6 - $4 = $2 per week. Since April has $150 more than Janelle initially, it will take them $150 / $2 = 75 weeks for their savings to be equal. Therefore, they will have the same amount of money in 75 weeks.
10.
Janelle has $20 and is saving $6 per week. April has $150 and is spending $4 per week. How much money will they both have when they have the same amount of money at the same time?
For your answer, fill in the blank below.
They will both have ______.
Explanation
In order to find out when Janelle and April will have the same amount of money, we need to set up an equation. Let x represent the number of weeks it takes for them to have the same amount of money. Janelle will have $20 + $6x and April will have $150 - $4x. We need to find the value of x when they have the same amount of money. Setting up the equation: $20 + $6x = $150 - $4x. Simplifying the equation, we get $10x = $130. Dividing both sides by $10, we get x = 13. Therefore, after 13 weeks, both Janelle and April will have $98.
11.
Sam and Hector are gaining weight for football season. Sam weighs 205 pounds and is gaining two pounds per week. Hector weighs 195 pounds but is gaining three pounds per week. In how many weeks will they both weigh the same amount?
For your answer, fill in the blank below.
They will have the same weight in ______.
Explanation
Since Sam is gaining 2 pounds per week and Hector is gaining 3 pounds per week, the difference in their weight gain per week is 1 pound. Therefore, it will take 10 weeks for them to both weigh the same amount, as Sam will have gained 20 pounds (2 pounds per week x 10 weeks) and Hector will have gained 30 pounds (3 pounds per week x 10 weeks), resulting in them both weighing 225 pounds.
12.
Sam and Hector are gaining weight for football season. Sam weighs 205 pounds and is gaining two pounds per week. Hector weighs 195 pounds but is gaining three pounds per week. How much will they both weigh when they are the same weight at the same time?
For your answer, fill in the blank below.
They will both weigh ______.
Explanation
Sam is gaining 2 pounds per week and Hector is gaining 3 pounds per week. In order for them to weigh the same, Sam needs to gain 20 pounds (since he currently weighs 205 pounds and Hector weighs 195 pounds). Since Sam is gaining 2 pounds per week, it will take him 10 weeks to gain 20 pounds. In that same time, Hector will have gained 30 pounds (since he is gaining 3 pounds per week). Therefore, they will both weigh 225 pounds when they are the same weight at the same time.
13.
Solve for x.
Explanation
The given equation is x = -1, -1, 1x = -1. This means that the value of x is equal to -1, and also equal to -1 when multiplied by 1. Therefore, the solution for x is x = -1.
14.
Solve for x.
Explanation
The given equation is x=13. This means that the value of x is equal to 13. Therefore, the correct answer is x=13. The additional answers mentioned, 13 and 1x=13, are redundant and do not provide any new information.
15.
Solve for x.
Explanation
The given equations are all equivalent and represent the value of x as 1.5 or 3/2.
16.
Solve for x.
Explanation
The given equation is x = 3. This means that the value of x is equal to 3. Additionally, the equation 3 is provided separately, indicating that 3 is also a possible value for x. Lastly, the equation 1x = 3 suggests that x multiplied by 1 is equal to 3, which again confirms that x = 3 is the correct answer.
17.
Solve for x.
Explanation
The given equation is "x=2, 2, 1x=2". This can be interpreted as three separate equations: x=2, 2, and 1x=2. The first equation states that x is equal to 2. The second equation states that 2 is equal to 2, which is true. The third equation states that 1 times x is equal to 2, which simplifies to x=2. Therefore, the solution to the equation is x=2.
18.
Solve for y:
5 + 2(x + y) = 11
Explanation
The given equation is 5 + 2(x + y) = 11. To solve for y, we need to isolate y on one side of the equation.
First, distribute the 2 to both terms inside the parentheses:
5 + 2x + 2y = 11
Next, subtract 5 from both sides of the equation:
2x + 2y = 6
Now, subtract 2x from both sides to isolate y:
2y = -2x + 6
Finally, divide both sides by 2 to solve for y:
y = -x + 3
19.
Solve for x:
x(2x − 1) = 2x^{2} + 5x − 12
Explanation
The given equation is a quadratic equation. To solve it, we can start by expanding the left side of the equation: x(2x - 1) = 2x^2 + 5x - 12. This simplifies to 2x^2 - x = 2x^2 + 5x - 12. By subtracting 2x^2 from both sides and combining like terms, we get -x = 5x - 12. Next, we can add x to both sides to eliminate the negative sign: 0 = 6x - 12. By adding 12 to both sides and dividing by 6, we find that x = 2. Therefore, the solution to the equation is x = 2.
20.
Solve for x:
4x(x + 1) = (2x − 3)(2x + 5)
Explanation
The given equation is a quadratic equation in the form of 4x(x + 1) = (2x - 3)(2x + 5). By expanding both sides of the equation, we get 4x^2 + 4x = 4x^2 - x - 15. Simplifying further, we get 4x = -x - 15. Rearranging the terms, we have 5x = -15. Dividing both sides by 5, we find that x = -3. However, substituting x = -3 back into the original equation, we get 0 = 0, which means that the equation is always true and has an infinite number of solutions. Therefore, the correct answer is that there is no solution.
21.
Give the equation of the line in y = mx+b form.
Explanation
The given equation of the line, y=2x-2, is already in the y=mx+b form. The equation represents a line with a slope of 2 and a y-intercept of -2. The other two options, y=2x+-2 and y=2x+(-2), are equivalent to the correct answer and also represent lines with the same slope and y-intercept.
22.
Give the equation of the line in y = mx+b form.
Explanation
The equation y = mx + b represents a line in slope-intercept form, where m is the slope of the line and b is the y-intercept. In this case, the equation y = -2x - 2 is already in the correct form, with m = -2 (the slope) and b = -2 (the y-intercept). Therefore, the given equation y = -2x - 2 is the correct answer. The other options y = -2x + (-2) and y = -2x + -2 are just alternative ways of writing the same equation.
23.
Give the equation of the line in y = mx+b form.
Explanation
The equation y = mx + b represents a line in slope-intercept form, where m is the slope of the line and b is the y-intercept. In the given equation y = -2x + 2, the slope is -2 and the y-intercept is 2. Therefore, the equation of the line is y = -2x + 2.
24.
Give the equation of the line in y = mx+b form.
Explanation
The equation y=2x+2 is in the form y=mx+b, where m represents the slope of the line and b represents the y-intercept. In this equation, the slope is 2 and the y-intercept is 2. This means that for every increase of 1 in x, the value of y will increase by 2. The line will intersect the y-axis at the point (0,2).
25.
What is the slope of the line ?
Explanation
The given answer states that the slope of the line is -2. In mathematics, slope represents the steepness of a line. A negative slope indicates that the line is decreasing as it moves from left to right. Therefore, a slope of -2 means that for every 1 unit increase in the horizontal direction, the line decreases by 2 units in the vertical direction. The answer also states that m is equal to -2, which is another way of representing the slope of the line.
26.
What is the y-intercept of the line ?
Explanation
The y-intercept of a line is the point where the line intersects the y-axis. In this case, the y-intercept is given as 8. This means that the line intersects the y-axis at the point (0,8), where the x-coordinate is 0 and the y-coordinate is 8.
27.
What is the slope of the line ?
Explanation
The slope of the line is -2. This is indicated by the value of m, which is also -2. The slope represents the steepness of the line, and a negative slope means that the line is decreasing as it moves from left to right. Therefore, the line has a slope of -2.
28.
What is the y-intercept of the line ?
Explanation
The y-intercept of a line represents the point where the line intersects the y-axis. In this case, the y-intercept is given as 6. This means that the line passes through the point (0,6) on the coordinate plane. The value of b is also given as 6, which indicates that the equation of the line can be written as y = 6x + 6.
29.
What is the x-intercept of the line ?
Explanation
The x-intercept of a line represents the point at which the line intersects the x-axis. In this case, the x-intercept is given as 3, which means that the line crosses the x-axis at the point (3,0).
30.
What is the x-intercept of the line ?
Explanation
The x-intercept of a line is the point where the line crosses the x-axis. In this case, the x-intercept is given as (4,0), which means that the line crosses the x-axis at the point (4,0). Therefore, the x-intercept of the line is 4.
31.
Which property is illustrated below?
8 + (3 + 6) = (8 + 3) + 6
Correct Answer
C. Associative property of addition
Explanation
The given equation demonstrates the associative property of addition. This property states that the grouping of numbers being added does not affect the sum. In the equation, the numbers 3 and 6 are grouped together in parentheses, and the sum of 3 + 6 is added to 8. On the other side of the equation, the numbers 8 and 3 are grouped together in parentheses, and the sum of 8 + 3 is added to 6. Both sides of the equation yield the same result, showing that the grouping of the numbers does not change the sum.
32.
Which of the following is the graph of the equation y = 6?
Correct Answer
C.
Explanation
The graph of the equation y = 6 is a horizontal line that passes through the y-axis at the point (0, 6). This is because the equation y = 6 means that the y-coordinate of every point on the graph is always 6, regardless of the value of x. Therefore, the graph is a straight line that is parallel to the x-axis and is located 6 units above it.
33.
What is the reciprocal of
Correct Answer
B.
34.
Which, if any, is the first incorrect setp in the solution show below?
Solve: 3x+15=2x+35
Step 1: 5x+15=35
Step 2: 5x=20
Step 3: x=4
Correct Answer
D. Step 1
Explanation
Step 1 is the first incorrect step in the solution shown because the equation was incorrectly simplified. The correct step would be to subtract 15 from both sides of the equation, resulting in 3x = 20.
35.
If x and y are opposites, which of the following relationships is true?
Correct Answer
C. X + y = 0
Explanation
If x and y are opposites, it means that they have opposite signs. For example, if x is positive, then y is negative, and vice versa. When we add two numbers with opposite signs, the result is always zero. Therefore, the relationship x + y = 0 is true when x and y are opposites.
36.
Which value of x is a counterexample to the conditional statement below?
If = 25, then x = 5.
Correct Answer
B. X = - 5
Explanation
A counterexample is an example that disproves a statement. In this case, the conditional statement is "If a = 25, then x = 5." The counterexample would be a case where a = 25 but x does not equal 5. In the given options, x = -5 is the only value that does not satisfy the statement. Therefore, x = -5 is the correct answer as it contradicts the given conditional statement.
37.
Which statement is false?
Correct Answer
A. The order in which two whole numbers are subtracted does not affect the difference.
Explanation
The statement that the order in which two whole numbers are subtracted does not affect the difference is true. When subtracting two whole numbers, changing the order of the numbers being subtracted does not change the result. This is because subtraction is not commutative, meaning the order matters. For example, if we subtract 5 from 10, we get a difference of 5. If we subtract 10 from 5, we get a difference of -5. Therefore, the statement is false.
38.
Which number does not have a reciprocal?
Correct Answer
B. 0
Explanation
The number 0 does not have a reciprocal because the reciprocal of a number is defined as 1 divided by the number, and dividing by 0 is undefined.
39.
What is the multiplicative inverse of ?
Correct Answer
D. 2
Explanation
The multiplicative inverse of a number is the number that, when multiplied by the original number, equals 1. In this case, the multiplicative inverse of 2 would be 1/2, because 2 multiplied by 1/2 equals 1.
40.
Which of the following points is the point of intersection of the lines of:
y = 2x - 3 and y = -x + 3?
Correct Answer
B. (2, 1)
Explanation
The point of intersection of two lines can be found by solving the system of equations formed by the two lines. In this case, the system of equations is y = 2x - 3 and y = -x + 3. By substituting the second equation into the first equation, we get -x + 3 = 2x - 3. Simplifying this equation, we find 3x = 6, which gives x = 2. Plugging this value of x into either of the original equations, we find y = 1. Therefore, the point of intersection is (2, 1).