Solve the system of linear equations by graphing. Round the solution to the nearest tenth.
[tex]\begin{array}{l}
y=-0.25 x+4.7 \\
y=4.9 x-1.64
\end{array}[/tex]
The approximate solution to the system is ( $\square$ , $\square$ ).
Answer (1)
Set the two equations equal to each other: − 0.25 x + 4.7 = 4.9 x − 1.64 .
Solve for x: x = 5.15 6.34 ≈ 1.2 .
Substitute x into one of the equations to solve for y: y = − 0.25 ( 1.2 ) + 4.7 = 4.4 .
The approximate solution to the system is ( 1.2 , 4.4 ) .
Explanation
Understanding the Problem We are given a system of two linear equations:
y = − 0.25 x + 4.7 y = 4.9 x − 1.64
We need to find the solution (x, y) to this system, rounded to the nearest tenth.
Setting the Equations Equal To solve the system of equations, we can set the two equations equal to each other:
− 0.25 x + 4.7 = 4.9 x − 1.64
Isolating x Now, we solve for x. Add 0.25 x to both sides and add 1.64 to both sides:
4.7 + 1.64 = 4.9 x + 0.25 x
This simplifies to:
6.34 = 5.15 x
Calculating x Divide both sides by 5.15 to find x:
x = 5.15 6.34 ≈ 1.231
Rounding to the nearest tenth, we get x ≈ 1.2 .
Substituting x to Find y Substitute the value of x into either of the original equations to solve for y. Let's use the first equation:
y = − 0.25 x + 4.7
y = − 0.25 ( 1.2 ) + 4.7
Calculating y Calculate the value of y:
y = − 0.3 + 4.7 = 4.4
Final Answer Therefore, the approximate solution to the system of equations is ( 1.2 , 4.4 ) .
Examples
Systems of linear equations are used in various real-world applications. For instance, they can model supply and demand in economics, where the intersection point represents the equilibrium price and quantity. In engineering, they can be used to analyze electrical circuits or structural stability. In everyday life, you might use them to determine the break-even point for a business venture or to optimize resource allocation.
