Fran graphs the equations $y=\frac{7}{4} x^2-2$ and $y=-0.5 x+4$. Her graph is shown below.
Which value is a solution of $\frac{7}{4} x^2-2=-0.5 x+4$?
$x=-2$
Answer (1)
Substitute x = − 2 into the equation 4 7 x 2 − 2 = − 0.5 x + 4 .
Evaluate the left-hand side: 4 7 ( − 2 ) 2 − 2 = 5 .
Evaluate the right-hand side: − 0.5 ( − 2 ) + 4 = 5 .
Since both sides are equal, x = − 2 is a solution: T r u e
Explanation
Checking the Solution We are given the equation 4 7 x 2 − 2 = − 0.5 x + 4 and asked to determine if x = − 2 is a solution. To check if x = − 2 is a solution, we substitute x = − 2 into the equation and see if both sides are equal.
Evaluating the LHS Substitute x = − 2 into the left-hand side (LHS) of the equation: 4 7 x 2 − 2 = 4 7 ( − 2 ) 2 − 2
Simplifying the LHS Evaluate the LHS: 4 7 ( − 2 ) 2 − 2 = 4 7 ( 4 ) − 2 = 7 − 2 = 5
Evaluating the RHS Substitute x = − 2 into the right-hand side (RHS) of the equation: − 0.5 x + 4 = − 0.5 ( − 2 ) + 4
Simplifying the RHS Evaluate the RHS: − 0.5 ( − 2 ) + 4 = 1 + 4 = 5
Conclusion Since the LHS equals the RHS when x = − 2 , then x = − 2 is a solution to the equation 4 7 x 2 − 2 = − 0.5 x + 4 .
Examples
Understanding how to solve equations is crucial in many real-world applications. For instance, engineers use equations to model the behavior of circuits and predict their performance. By substituting different values into the equation, they can determine the optimal components to use. Similarly, in economics, equations are used to model supply and demand curves. By finding the point where the two curves intersect, economists can determine the equilibrium price and quantity of a product. This skill is also essential in fields like physics, where equations describe the motion of objects, and in computer science, where algorithms are designed and analyzed using mathematical equations.
