Set up an equation to find x, then find the perimeter of an equilateral triangle with sides of length 3x and 7x-12.
An equilateral triangle is one with all 3 sides of the same length. If all sides are the same length, then we can set up an equation stating that the side 3x is the same length as the side 7x-12, so 3x = 7x-12. To solve the equation for x, we need to combine terms that are alike. Since 3x and 7x are on opposite sides of the equal sign, we need to use the additive property of equality. It makes sense in this case to add -7x to both sides of the equation creating: -7x + 3x = -7x + 7x - 12. Now we simplify: -7x + 3x is -4x, and -7x + 7x is zero. Our equation is now -4x=0-12 or -4x=-12. Now we use the Division Property of Equality to divide both sides by -4 to determine the value of x. -4x divided by -4 is 1x or x, and -12 divided by -4 is 3. Therefore, x has a value of 3. We use x to find the lengths of the sides of the triangle: 3x becomes 3(3) which is 9. 7x-12 becomes 7(3)-12 or 21-12 which is also 9. *That's good because we are supposed to be working with an equilateral triangle! The perimeter is the sum of the sides, so an equilateral triangle with sides of length 9 would have a perimeter of 27 units.
Describe or show why 6 divided by 1/3 = 18 and not 2.
We can think of division as taking the amount we are given (the dividend - in this case "6") and dividing it into parts that are the size of the divisor (in this case "1/3) and then counting the number of groups that are formed to determine the quotient (in this case "18"). If we take 6 whole candy bars and cut each one into three equal pieces, we are dividing them into thirds. The result would be 18 pieces that are each 1/3 of a candy bar in size. Therefore, 18 is the quotient of 6 and 1/3. 2 is the quotient to the problem 6 divided by 3. Using the same logic as above, we would take the amount we are given (6) and divide it into equal groups that are each "3" in size. Then we count the number of groups created, which would be 2.
Describe 3 ways to write the equation in slope-intercept form for a line containing the points (-2,4) and (6,-8).
To write an equation in slope-intercept form (y=mx+b), we need to find the slope (m) and the y-intercept (0,b). One method involves graphing the given points on a coordinate plane and drawing the line that contains them. From the graph, we can count the "rise over run" (vertical distance between the points over the horizontal distance between the points) to determine the slope of the line (-3/2). We can also find the y-intercept on the graph by locating the point where our line crosses the y-axis (0,1). We now fill in the slope and intercept to write the equation: y = -3/2 x + 1. It is wise to check our work to make sure both points satisfy the equation: 4 = -3/2(-2) + 1 and -8 = -3/2(6) + 1. Not all lines contain a y-intercept that is easily determined by looking at its graph. We have other methods of discovering slope and y-intercept without a graph. Since the slope is a number that describes the rate of change in y over the change in x, we can find slope easily when we are given the coordinates of two points on the line. We subtract the values of y from the ordered pairs: 4- -8 and the values of x: -2 -6 to find that the slope is 12/-8, which simplifies to -3/2. Now that we know the slope, we can use one of the ordered pairs and the slope to substitute into the slope-intercept equation and solve it to find the missing variable "b" - which is the y-intercept. Let's use the first ordered pair (-2,4). y = mx + b becomes 4 = -3/2(-2) + b and we solve for "b." Multiply -3/2(-2) and 4 = 3 + b, so b = 1. We now have the information needed to write the equation: y = -3/2x + 1. The third method involves finding the slope just as we did in the second method above. We can first write the equation in point-slope form: y-y_1= m(x-x_1) to begin. Again, it does not matter which point we use, so this time we will use (-2,4). In point-slope form this would be: y - 4 = (-3/2)(x - -2). Then we manipulate the equation to re-write it in slope-intercept form. It first becomes y-4=-3/2x - 3 using the distributive property. Then adding 4 to both sides yields the equation in slope-intercept form: y = -3/2x + 1