Select the equations that are linear equations. (8 points)
sqrt(x) = 2 + 9x
3x - 2 = 7
2x + 7x = -2
2x^5 -2x = 6
The linear equations are:
3x - 2 = 7
2x + 7x = -2
In order to determine which equations are linear equations, we need to understand what defines a linear equation. A linear equation is an equation in which the highest power of the variable is 1 (the variable is not raised to any exponent other than 1) and there are no variables within functions like square roots or exponents.
Let's analyze each of the given equations:
1. sqrt(x) = 2 + 9x
This equation involves a square root, so it is not a linear equation.
2. 3x - 2 = 7
This equation has a variable raised to the first power, and there are no functions or exponents involved besides the variable itself. Therefore, it is a linear equation.
3. 2x + 7x = -2
This equation contains variables raised to the first power, and addition and multiplication operations. It is also a linear equation.
4. 2x^5 - 2x = 6
This equation involves a variable raised to the fifth power (x^5), which exceeds the highest power of 1. Thus, it is not a linear equation.
Therefore, the linear equations from the given list are:
- 3x - 2 = 7
- 2x + 7x = -2
To determine which equations are linear equations, we need to identify equations that can be written in the form y = mx + b, where m and b are constants, and x is the variable.
Let’s go through each equation and check if it fits the criteria:
1. sqrt(x) = 2 + 9x:
This equation is not linear because the variable x appears within a square root function, which is not a linear relationship.
2. 3x - 2 = 7:
This equation is linear because it can be rearranged to the form y = mx + b. By moving the -2 to the right side of the equation, we get 3x = 7 + 2, which simplifies to y = 3x + 2.
3. 2x + 7x = -2:
This equation is linear because we can combine like terms on the left side of the equation to get 9x = -2. By dividing both sides by 9, we get y = x/9.
4. 2x^5 - 2x = 6:
This equation is not linear because it contains a term with x raised to the power of 5. Linear equations only contain terms with x raised to the power of 1 (or no exponent at all).
Therefore, the linear equations out of the given options are:
3x - 2 = 7
2x + 7x = -2