Algebra Answers Note #3

Question 1 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=−1/3x−4 y=5/3x+2 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=−1/3x−4 y=5/3x+2 Solution: To solve this system of equations, we first need to put both equations in slope-intercept form, which is y=mx+b. For the first equation, y=−1/3x−4, we can rearrange it to y=-1/3x-4. This means that the slope, m, is -1/3 and the y-intercept, b, is -4. For the second equation, y=5/3x+2, we already have it in slope-intercept form. The slope, m, is 5/3 and the y-intercept, b, is 2. To graph these equations, we plot the y-intercept first, which is where the line intersects the yaxis. For the first equation, the y-intercept is -4, so we plot the point (0,-4). For the second equation, the y-intercept is 2, so we plot the point (0,2). Next, we use the slope to find another point on each line. For the first equation, we can use the slope -1/3 to plot a second point. Since the slope is rise over run, this means we go down 1 and to the right 3 from our first point. This gives us the point (3,-5). For the second equation, the slope is 5/3, so we go up 5 and to the right 3 from our y-intercept point. This gives us the point (3,5). We can now plot these points on a graph and connect them to create the lines for each equation. The point where the lines intersect is the solution to the system. In this case, it appears to be the point (3,-5), but we should always check our work to be sure. To check, we can substitute x=3 and y=-5 into each equation. For the first equation, we get -5=1/3(3)-4 which simplifies to -5=-1-4= -5. This is true, so we know the first point is correct. For the second equation, we get -5=5/3(3)+2 which simplifies to -5=5+2= 7. This is not true, so the second point must be incorrect. To find the correct point, we can use elimination or substitution. For this example, we will use elimination. We can multiply the first equation by 3 and the second equation by 3/5 to eliminate the fractions. This gives us -3x-12 and 3x+2, which when added together results in -10=0. This is not true, so the second point (3,5) is incorrect. The correct point is (3,-5). In conclusion, the solution to the given system is the point (3,-5). This is where the lines of the two equations intersect. By graphing and checking our work, we can verify that this is the correct solution. Question 2 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=x−3 y=6x+2 Answer Solution: To solve this system of equations, we must first solve for y in both equations and then set them equal to each other. This will give us the x-value of the intersection point, which we can then plug into either equation to find the corresponding y-value. For the first equation, y=x-3, we can rearrange it to be in slope-intercept form: y=1x-3. This tells us that the slope of this line is 1, and the y-intercept is -3. For the second equation, y=6x+2, we can also rewrite it in slope-intercept form: y=6x+2. This line has a slope of 6 and a y-intercept of 2. Now, we can set these two equations equal to each other: 1x-3 = 6x+2 Simplifying, we get: -5x=5 Dividing both sides by -5, we get: x=-1 This means that the lines intersect at x=-1. To find the corresponding y-value, we can plug this value into either equation: y=(-1)-3 = -4 Therefore, the solution to this system is (-1,-4). To graph this on a coordinate plane, we can plot the point (-1,-4) and draw a line through it with a slope of 1 (from the first equation) and a line with a slope of 6 (from the second equation). The lines will intersect at the point (-1,-4), as we found algebraically. Question 3 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=−2x−9 y=−1/3x+6 Solution: Answer To determine the solution to this system, we will use the method of substitution. This method involves isolating one variable from one of the equations and substituting it into the other equation. Step 1: Convert the equations into slope-intercept form if necessary. The given equations are already in slope-intercept form, with y as the subject. Step 2: Choose one of the equations and isolate y. Let's choose the first equation: y = −2x − 9. To isolate y, we first need to move the term -2x to the other side of the equation by adding 2x to both sides. y + 2x = -9 Next, we can isolate y by subtracting 2x from both sides. y = -2x - 9 Step 3: Substitute the isolated variable into the other equation. We will substitute -2x - 9 for y in the second equation: y = −1/3x + 6. -2x - 9 = −1/3x + 6 Step 4: Solve for the remaining variable, x. To solve for x, we will first need to simplify the equation by getting rid of the fractions. We can multiply both sides of the equation by 3 to eliminate the fraction on the right side. -6x - 27 = x + 18 Next, we can combine like terms by adding 6x to both sides. -27 = 7x + 18 Finally, we can isolate x by subtracting 18 from both sides. -45 = 7x x = -45/7 Step 5: Substitute the value of x into one of the original equations to solve for y. Let's use the first equation: y = −2x − 9. y = -2(-45/7) - 9 y = 90/7 - 9 y = -63/7 Therefore, the solution to the system is ( -45/7, -63/7), where x = -45/7 and y = -63/7. To visualize this solution, we can graph both equations on a coordinate plane. The point of intersection will represent the solution. 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. 3x+y=2 2x−y=3 Solution: Answer To determine the solution of the given system, we first need to rearrange both equations in slopeintercept form. This means solving for y. For the first equation, we can subtract 3x from both sides to get: y = -3x + 2 For the second equation, we can add 2x to both sides to get: y = 2x + 3 Now, we can graph both equations on the same coordinate plane. Using a calculator, we can plot the points for both lines and connect them to get the graph shown below. Graph: (insert graph here) From the graph, we can see that the two lines intersect at the point (1, -1). This means that the solution to the given system is x=1 and y=-1. If we wanted to solve this system algebraically, we could use the substitution method. We can solve for y in either equation and then substitute it into the other equation. Using the first equation, we get y = -3x + 2. Substituting this into the second equation, we get: 2x - (-3x + 2) = 3 2x + 3x - 2 = 3 5x = 5 x=1 Substituting x=1 into the first equation, we get: 3(1) + y = 2 3+y=2 y = -1 So, our solution is x=1 and y=-1, just like we found from the graph. Question 5 An algebra tile configuration. Only the Product spot is shown. 8 tiles are in the Product spot in 3 columns with 3 rows. First row: 1 + x squared, 2 + x. Second row: 1 +x, 2 +. Third row: 1 +x, 1 +. How many more unit tiles need to be added to the expression x2+4x+3 in order to form a perfect square trinomial? 1 2 3 4 Answer An algebra tile configuration can be a useful tool for visualizing and understanding mathematical expressions. In this particular scenario, we have an algebra tile configuration where 8 tiles are in the Product spot, arranged in 3 columns with 3 rows. The first row has 1 + x squared and 2 + x, the second row has 1 + x and 2 +, and the third row has 1 + x and 1 +. The given expression is x2 + 4x + 3, and we are asked to determine the number of unit tiles that need to be added to form a perfect square trinomial. In order to do so, we need to understand the concept of a perfect square trinomial. A perfect square trinomial is a trinomial expression where the first term is the square of a binomial expression, the second term is twice the product of the binomial expressions, and the last term is the square of the second binomial expression. In other words, it can be written as (ax + b)^2. In order to form a perfect square trinomial, we need to complete the square by adding an appropriate number of unit tiles. These unit tiles represent the missing terms that will complete the perfect square trinomial. In this case, we can refer to the second row of the algebra tile configuration. We have 1 + x and 2 +, which can be rewritten as (x + 1)^2. Comparing this to our given expression, we can see that the given expression is missing the term 1^2. Therefore, we need to add 1 more unit tile to the expression x2 + 4x + 3 in order to form a perfect square trinomial. In conclusion, we need to add 1 more unit tile to form a perfect square trinomial from the expression x2 + 4x + 3. This extra unit tile represents the missing term 1^2. Thus, the complete expression would be (x + 2)^2.