6. The perimeter of this shape is 18 inches. What is the unknown side length?
[tex]$\begin{array}{l}
x=18-12= \
x=6\end{array}$[/tex]
7. Find the slope and [tex]$y$[/tex]-intercept of each equation.
A) [tex]$-5-y=-3 x$[/tex]
B) [tex]$5 y+10=-2 x$[/tex]
Answer (2)
Problem 6: The unknown side length is found to be 6 inches by verifying the perimeter equation. Problem 7A: The equation − 5 − y = − 3 x is converted to slope-intercept form, revealing a slope of 3 and a y-intercept of -5. Problem 7B: The equation 5 y + 10 = − 2 x is also converted to slope-intercept form, showing a slope of − 5 2 and a y-intercept of -2. The final answers are: Problem 6: 6 , Problem 7A: slope 3 , y-intercept − 5 , Problem 7B: slope − 5 2 , y-intercept − 2 .
Explanation
Problem Overview We are given two separate problems. The first problem involves finding an unknown side length given the perimeter of a shape. The second problem involves finding the slope and y-intercept of two linear equations. We will solve each problem separately.
Verifying the Unknown Side Length For the first problem, we are given that the perimeter of a shape is 18 inches and that x = 18 − 12 = 6 . We need to verify if x = 6 is the correct unknown side length. The equation suggests that 12 inches represents the sum of the known sides. If x = 6 is the unknown side, then the sum of all sides is 12 + 6 = 18 inches, which matches the given perimeter. Therefore, the solution is correct.
Finding Slope and Y-intercept for Equation A For the second problem, part A, we need to find the slope and y-intercept of the equation − 5 − y = − 3 x . To do this, we rewrite the equation in slope-intercept form ( y = m x + b ), where m is the slope and b is the y-intercept. Adding 5 to both sides gives − y = − 3 x + 5 . Multiplying both sides by -1 gives y = 3 x − 5 . Therefore, the slope is m = 3 and the y-intercept is b = − 5 .
Finding Slope and Y-intercept for Equation B For the second problem, part B, we need to find the slope and y-intercept of the equation 5 y + 10 = − 2 x . To do this, we rewrite the equation in slope-intercept form ( y = m x + b ). Subtracting 10 from both sides gives 5 y = − 2 x − 10 . Dividing both sides by 5 gives y = − 5 2 x − 2 . Therefore, the slope is m = − 5 2 and the y-intercept is b = − 2 .
Final Answers In summary:
Problem 6: The unknown side length is 6 inches. Problem 7A: The slope is 3 and the y-intercept is -5. Problem 7B: The slope is - 5 2 and the y-intercept is -2.
Examples
Understanding perimeters and linear equations is crucial in various real-world applications. For instance, when designing a garden, knowing the perimeter helps determine the amount of fencing needed. Similarly, understanding slope and y-intercept is useful in analyzing linear relationships, such as the cost of a service based on usage, where the slope represents the rate of change and the y-intercept represents the initial fee. These concepts are fundamental in fields like architecture, engineering, and economics, enabling professionals to make informed decisions and solve practical problems.
The unknown side length is 6 inches. The slope of the equation − 5 − y = − 3 x is 3 with a y-intercept of -5, while for 5 y + 10 = − 2 x , the slope is -\frac{2}{5} with a y-intercept of -2.
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