Which linear equation can be derived from this proportion?
[tex]\frac{12}{x+8}=\frac{7}{x+1}[/tex]
A. [tex]$12 x+96=7 x+7$[/tex]
B. [tex]$12 x+1=7 x+8$[/tex]
C. [tex]$12 x+12=7 x+56$[/tex]
D. [tex]$12 x+8=7 x+1$[/tex]
Answer (1)
Cross-multiply the given proportion: 12 ( x + 1 ) = 7 ( x + 8 ) .
Expand both sides of the equation: 12 x + 12 = 7 x + 56 .
Compare the expanded equation with the given options.
The correct linear equation is: 12 x + 12 = 7 x + 56
Explanation
Understanding the Problem We are given the proportion x + 8 12 = x + 1 7 and asked to find the equivalent linear equation from the given choices. To do this, we will cross-multiply to eliminate the fractions and then simplify the resulting equation.
Cross-Multiplication Cross-multiply the proportion: 12 ( x + 1 ) = 7 ( x + 8 ) This step eliminates the fractions by multiplying both sides by ( x + 8 ) ( x + 1 ) .
Expanding the Equation Expand both sides of the equation using the distributive property: 12 x + 12 = 7 x + 56 This simplifies the equation into a linear form.
Identifying the Correct Option Compare the resulting equation 12 x + 12 = 7 x + 56 with the given options: A. 12 x + 96 = 7 x + 7 B. 12 x + 1 = 7 x + 8 C. 12 x + 12 = 7 x + 56 D. 12 x + 8 = 7 x + 1 We see that option C matches our derived equation.
Final Answer Therefore, the correct linear equation derived from the given proportion is 12 x + 12 = 7 x + 56 .
Examples
Proportions and linear equations are used in various real-life scenarios, such as scaling recipes, calculating distances on maps, and determining the correct mixture ratios in construction or chemistry. For instance, if you need to scale a recipe that feeds 4 people to feed 12 people, you would use proportions to adjust the amount of each ingredient. Similarly, in construction, if a blueprint uses a scale of 1 inch = 5 feet, you would use a proportion to determine the actual dimensions of a room or building feature.
