Write the equation in the standard linear equation form $(A x+B y=C)$ if possible. On the blank, write Yes if it is linear. If the answer is No, explain why.
1. $\frac{y}{3}+x=5$
Explain if No.
2. $3 x+5^y=8$
Explain if No.
3. $x+9 y^2=1$
Explain if No.
4. $1+\frac{7}{x}=y$
Explain if No.
5. $7+x=2 y$
Explain if No.
Which table of values represents the equation $y=2 x+5$ ?
A.
| x | y |
| --- | --- |
| -8 | -11 |
| -7 | -9 |
| -6 | -7 |
| -5 | -5 |
| -4 | -3 |
B.
| x | y |
| --- | --- |
| -3 | -1 |
| -1 | 3 |
| 1 | 7 |
| 3 | 11 |
| 5 | 15 |
C.
| x | y |
| --- | --- |
| 0 | 9 |
| 1 | 7 |
| 2 | 5 |
| 3 | 3 |
| 4 | 1 |
D.
| x | y |
| --- | --- |
| -5 | -5 |
| -4 | -3 |
| -3 | -1 |
| -2 | 1 |
| -1 | 3 |
Answer (2)
Rewrite the first equation 3 y + x = 5 in standard form as 3 x + y = 15 .
Identify that 3 x + 5 y = 8 and x + 9 y 2 = 1 are not linear due to the exponential term and the squared term, respectively.
Recognize that 1 + x 7 = y is not linear because of the term x 7 .
Express 7 + x = 2 y in standard form as x − 2 y = − 7 and determine that Table D represents the equation y = 2 x + 5 .
The final answer is that the first equation is 3 x + y = 15 , the second and third are not linear, the fourth is not linear, the fifth is x − 2 y = − 7 , and the correct table is D. D
Explanation
Problem Analysis Let's analyze each equation to determine if it's linear and, if so, convert it to standard form ( A x + B y = C ) . We'll also identify the correct table of values for the given linear equation.
Equation 1
3 y + x = 5 can be rewritten as x + 3 y = 5 . Multiplying the entire equation by 3 to eliminate the fraction gives 3 x + y = 15 . This is in the standard form.
Equation 2
3 x + 5 y = 8 . The presence of the term 5 y (where y is in the exponent) indicates that this equation is not linear.
Equation 3
x + 9 y 2 = 1 . The presence of the term 9 y 2 indicates that this equation is not linear because the variable y is squared.
Equation 4
1 + x 7 = y . The presence of the term x 7 indicates that this equation is not linear because x is in the denominator.
Equation 5
7 + x = 2 y can be rearranged to x − 2 y = − 7 . This is in the standard form.
Table Analysis Now, let's find the table of values that represents the equation y = 2 x + 5 . We need to check each table to see which one satisfies the equation.
Checking the Tables
Table A:
If x = − 8 , then y = 2 ( − 8 ) + 5 = − 16 + 5 = − 11 . The table shows y = 13 , which is incorrect.
If x = − 7 , then y = 2 ( − 7 ) + 5 = − 14 + 5 = − 9 . The table shows y = 11 , which is incorrect.
If x = − 6 , then y = 2 ( − 6 ) + 5 = − 12 + 5 = − 7 . The table shows y = 9 , which is incorrect.
If x = − 5 , then y = 2 ( − 5 ) + 5 = − 10 + 5 = − 5 . The table shows y = 7 , which is incorrect.
If x = − 4 , then y = 2 ( − 4 ) + 5 = − 8 + 5 = − 3 . The table shows y = 5 , which is incorrect.
Table B:
If x = − 3 , then y = 2 ( − 3 ) + 5 = − 6 + 5 = − 1 . The table shows y = − 3 , which is incorrect.
If x = 5 , then y = 2 ( 5 ) + 5 = 10 + 5 = 15 . The table shows y = − 2 , which is incorrect.
If x = 7 , then y = 2 ( 7 ) + 5 = 14 + 5 = 19 . The table shows y = − 1 , which is incorrect.
If x = 9 , then y = 2 ( 9 ) + 5 = 18 + 5 = 23 . The table shows y = 0 , which is incorrect.
If x = 11 , then y = 2 ( 11 ) + 5 = 22 + 5 = 27 . The table shows y = 1 , which is incorrect.
Table C:
If x = 0 , then y = 2 ( 0 ) + 5 = 0 + 5 = 5 . The table shows y = 9 , which is incorrect.
If x = 1 , then y = 2 ( 1 ) + 5 = 2 + 5 = 7 . The table shows y = 7 , which is correct.
If x = 2 , then y = 2 ( 2 ) + 5 = 4 + 5 = 9 . The table shows y = 5 , which is incorrect.
If x = 3 , then y = 2 ( 3 ) + 5 = 6 + 5 = 11 . The table shows y = 3 , which is incorrect.
If x = 4 , then y = 2 ( 4 ) + 5 = 8 + 5 = 13 . The table shows y = 1 , which is incorrect.
Table D:
If x = − 5 , then y = 2 ( − 5 ) + 5 = − 10 + 5 = − 5 . The table shows y = − 5 , which is correct.
If x = − 4 , then y = 2 ( − 4 ) + 5 = − 8 + 5 = − 3 . The table shows y = − 3 , which is correct.
If x = − 3 , then y = 2 ( − 3 ) + 5 = − 6 + 5 = − 1 . The table shows y = − 1 , which is correct.
If x = − 2 , then y = 2 ( − 2 ) + 5 = − 4 + 5 = 1 . The table shows y = 1 , which is correct.
If x = − 1 , then y = 2 ( − 1 ) + 5 = − 2 + 5 = 3 . The table shows y = 3 , which is correct.
Final Answer Therefore, Table D correctly represents the equation y = 2 x + 5 .
Examples
Linear equations are used in various real-life scenarios, such as calculating the cost of items, determining the distance traveled at a constant speed, or modeling simple relationships between two variables. For example, if you are buying apples at $2 per apple and have a coupon for 5 o ff , t h e t o t a l cos t c anb ere p rese n t e d b y t h e l in e a re q u a t i o n y = 2x - 5 , w h ere x i s t h e n u mb ero f a ppl es an d y$ is the total cost. Understanding linear equations helps in making informed decisions and predictions in everyday situations.
The equations were analyzed to determine linearity, resulting in two being linear and three not. Additionally, Table A correctly represents the linear equation y = 2 x + 5 .
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