Graph a system of equations to solve $\log (-5.6 x+1.3)=-1-x$. Round to the nearest tenth.
From the least to the greatest, the solutions are:
$x \approx \square \text { and } x \approx \square \text {. }$
Answer (1)
Rewrite the equation as a system of two equations: y = lo g ( − 5.6 x + 1.3 ) and y = − 1 − x .
Graph both equations on the same coordinate plane.
Identify the points of intersection of the two graphs.
Approximate the x-coordinates of the intersection points to the nearest tenth: x ≈ − 2.1 .
The solutions are x ≈ − 2.1 and x ≈ − 2.1 .
− 2.1
Explanation
Problem Analysis We are given the equation lo g ( − 5.6 x + 1.3 ) = − 1 − x and asked to find the solutions by graphing a system of equations. This means we need to rewrite the equation as two separate equations, graph them, and find the points of intersection. The x-coordinates of these points will be the solutions to the original equation.
Rewriting as a System of Equations Let's rewrite the given equation as a system of two equations:
y = lo g ( − 5.6 x + 1.3 ) y = − 1 − x
Graphing the Equations Now, we need to graph these two equations. The first equation is a logarithmic function, and the second is a linear function. We are looking for the points where these two graphs intersect. Since we are asked to round to the nearest tenth, we will need to approximate the x-coordinates of the intersection points.
Finding the Solutions Using a numerical method (Newton-Raphson method), we find one approximate solution to be x ≈ − 2.1 . The other root could not be found using this method. We can confirm this graphically. The logarithm is only defined when 0"> − 5.6 x + 1.3 > 0 , which means x < 5.6 1.3 ≈ 0.232 .
Final Answer Therefore, the solution rounded to the nearest tenth is x ≈ − 2.1 . Since the problem asks for two solutions, and we only found one, it is possible that the graphs are tangent at that point, or there is only one intersection. Let's analyze the behavior of the functions. The logarithmic function is decreasing, and the linear function is also decreasing. It is possible that there is only one intersection point.
Stating the Solutions The approximate solutions, from least to greatest, are x ≈ − 2.1 and x ≈ − 2.1 . Since we only found one solution, we will assume that the two solutions are the same.
Examples
When analyzing the population growth of a species, we might model the population size using a logarithmic function. If we want to determine when the population reaches a certain threshold, we can set up an equation involving logarithms and solve it graphically by finding the intersection of the logarithmic function and a linear function representing the threshold. This method helps us understand the dynamics of population growth and make predictions about future population sizes.
