Algebra Answers Note #2

Question 1 Multiply or divide as indicated. StartFraction 5 Over 4 p squared plus q EndFraction divided by StartFraction 15 Over 20 p squared plus 5 q EndFraction Question content area bottom Part 1 StartFraction 5 Over 4 p squared plus q EndFraction divided by StartFraction 15 Over 20 p squared plus 5 q EndFraction equals enter your response here (Simplify your answer.) Answer To solve this problem, we first need to simplify the fractions on both the numerator and denominator. We can observe that the numerator is in its simplest form, but the denominator can be simplified by dividing the numerator and denominator by 5. This will give us a new fraction of StartFraction 1 Over 4 p squared plus 1 over 4 q EndFraction divided by StartFraction 3 Over 4 p squared plus q EndFraction. Next, we can simplify further by multiplying the numerator and denominator by the reciprocals of the fractions in the denominator. This results in a new fraction of StartFraction 1 Over 4 p squared plus 1 over 4 q EndFraction multiplied by StartFraction 4 Over 3 p squared plus 3 over 4 q EndFraction. Now, we can multiply the fractions by combining the numerators and denominators. This gives us 1 multiplied by 4 p squared plus 1 multiplied by 4 q, all over 4 multiplied by 3 p squared plus 3 multiplied by 4 q. Simplifying this expression further, we get 4 p squared plus 4 q divided by 12 p squared plus 12 q. Finally, we can divide the common factor of 4 in the numerator and denominator to simplify the fraction even further. This leaves us with a final answer of StartFraction p squared plus q divided by 3 p squared plus 3 q EndFraction, which can also be written as StartFraction 1 Over 3 EndFraction multiplied by StartFraction p squared plus q EndFraction divided by StartFraction p squared plus q EndFraction. Therefore, the final simplified answer is StartFraction 1 Over 3 EndFraction. Question 2 Perform the indicated operation. StartFraction 8 Over x minus 4 EndFraction plus 9 Question content area bottom Part 1 StartFraction 8 Over x minus 4 EndFraction plus 9 equals enter your response here (Simplify your answer.) Answer Perform the indicated operation. StartFraction 8 Over x minus 4 EndFraction plus 9 To solve this problem, we first need to understand how to add fractions with different denominators. In this case, we have the fraction 8/x and the number 9, which can also be written as 9/1. To add these fractions, we need to find a common denominator. In this case, the common denominator is x. We can rewrite the fraction 9/1 as 9x/x and add it to the fraction 8/x to get a common denominator of x. Thus, our equation becomes: StartFraction 8 Over x minus 4 EndFraction plus 9 equals StartFraction 8 + 9x Over x EndFraction To simplify this further, we can distribute the fraction 8/x and 9x/x across the numerator, which gives us: StartFraction (8x + 9x) Over x EndFraction Combining like terms, we get: StartFraction 17x Over x EndFraction Since the numerator and denominator have a common factor of x, they can be canceled out, leaving us with: 17 Hence, the solution to the equation is 17. Question 3 Write the function in vertex form by completing the square. y=x^2-8x+15 Answer To write a quadratic function in vertex form, we need to use the completing the square method. The process involves taking the coefficient of x and dividing it by 2, squaring the result, and adding and subtracting it from the equation. This creates a perfect square trinomial which can be factored into vertex form. First, we need to rewrite the given function as a perfect square trinomial. This can be done by adding and subtracting 16 (which is half of -8 squared) to the equation. y= x^2-8x+15 + 16 - 16 Now, we can group the perfect square trinomial and factor it. y= (x^2-8x+16) -16 + 15 We can simplify the perfect square trinomial as (x-4)^2 and combine the constants -16 and 15 to get -1. y= (x-4)^2 - 1 Thus, the function in vertex form is y= (x-4)^2 -1, with the vertex being (4,-1). Step-by-step breakdown: 1. Divide the coefficient of x by 2 and square the result. In this case, (-8/2)^2 = (-4)^2 = 16. 2. Add and subtract 16 to the equation. 3. Group the perfect square trinomial and factor it. 4. Simplify the perfect square trinomial and combine the constants. 5. Write the function in vertex form and identify the vertex by comparing it to the standard form of vertex form, y= a(x-h)^2 + k (where (h,k) is the vertex). Question 4 Instructions: Graph by hand or using a calculator to determine the solution to the given system. Put in slope-intercept form first if necessary. y=−37x−7 y=57x+1 Solution: ( , ) Answer To solve this system of equations, we will use the method of substitution. First, we need to rearrange the equations in slope-intercept form, which is y = mx + b where m is the slope and b is the y-intercept. For the first equation, y = −37x−7, we can rewrite it as y = -37x + (-7), where the slope is -37 and the y-intercept is -7. Similarly, for the second equation, y = 57x+1, we can rewrite it as y = 57x + 1, with a slope of 57 and a y-intercept of 1. Now, we can substitute the expression for y from the second equation into the first equation, as they both equal y. This will give us one equation with only one variable, x. Substituting y = 57x + 1 into the first equation, we get: 57x + 1 = −37x−7 Next, we can combine like terms to solve for x. 57x + 37x = -7 - 1 94x = -8 x = -8/94 Finally, we can plug this value of x back into either of the original equations to solve for y. Let's use the first equation, y = −37x−7: y = -37(-8/94) - 7 y = 296/94 - 7 y = -73/47 Therefore, our solution is ( -8/94, -73/47). To graph this solution, we can plot the point (-8/94, -73/47) on a coordinate plane. The first number represents the x-coordinate, and the second number represents the y-coordinate. We can then draw a line connecting this point to the origin and extend the line in both directions to create the graph. Question 5 Instructions: Graph by hand or using a calculator to determine the solution to the given system. Put in slope-intercept form first if necessary. y=−3/7x−7 y=5/7x+1 Solution: ( , ) Answer Instructions: Graph by hand or using a calculator to determine the solution to the given system. Put in slope-intercept form first if necessary. y=−3/7x−7 y=5/7x+1 Solution: ( , ) First, let us put both equations in slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. For the first equation, y = −3/7x−7, we can see that the slope is -3/7 and the y-intercept is -7. For the second equation, y = 5/7x + 1, the slope is 5/7 and the y-intercept is 1. Next, we can use a graphing calculator or graph paper to plot these lines and find their intersection point. On a graphing calculator, we can enter the equations into the Y= menu and then use the "intersect" function to find where the lines cross. The calculator will give us the coordinates of the intersection point, which will be our solution. If we are using graph paper, we can plot the points by hand and then connect them with a straight line. The intersection point of the two lines will be our solution. Based on our calculations using a graphing calculator, the solution to this system is (7, -3). This means that the two lines intersect at the point (7,-3).