Example Question - linear terms
Here are examples of questions we've helped users solve.
Solving an Equation with Exponential and Linear Terms
The equation in the image is \( e^{x} + 7x + 11 = 20 \). To solve for \( x \), follow these steps: 1. First, move constants to one side to isolate the \( x \)-terms: \[ e^{x} + 7x + 11 - 20 = 0 \] \[ e^{x} + 7x - 9 = 0 \] 2. This equation does not have a straightforward algebraic solution because it combines an exponential function and a linear term. To solve for \( x \), we typically use numerical methods such as the Newton-Raphson method or by graphically finding the intersection point. For a simple approximate solution, we can try a few guesses: - If \( x = 0 \), \( e^0 + 7(0) - 9 = 1 + 0 - 9 = -8 \), which is too low. - If \( x = 1 \), \( e^1 + 7(1) - 9 = e + 7 - 9 \). Given that \( e \approx 2.718 \), this is \( 2.718 + 7 - 9 \approx 0.718 \), which is too high. Since the function \( e^{x} + 7x - 9 \) is continuous, there will be a root between 0 and 1. To find the exact root, you would use a numerical solver or iterate with guesses refining towards the point where the function equals zero.
Solving Equations with Exponential and Linear Terms
The equation provided in the image is: e^x + 7x + 11 = 20 To solve it, we should try to isolate x. First, we'll subtract 11 from both sides to get the terms involving x by themselves: e^x + 7x + 11 - 11 = 20 - 11 e^x + 7x = 9 Now, we have an equation with both an exponential and a linear term in x. This is not a standard algebraic equation that can be solved through elementary algebraic manipulations. Instead, we would typically use numerical methods to approximate the value of x, such as the Newton-Raphson method or by using graphing techniques. For an exact algebraic solution, there is no simple method. You would likely need to use a computational tool or a graphing calculator to approximate the value of x by finding the intersection of the graph y = e^x + 7x with the horizontal line y = 9. If you need to solve this equation exactly, and if you have access to a calculator or a computer, you can plug the function f(x) = e^x + 7x - 9 into a root-finding algorithm to find the solution for x that makes the equation true.
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