Question

the method that involves plotting each equation on a coordinate plane and finding the intersection point is called ______.
a real - world application of systems of equations where you allocate hours to various activities is called ______ management.
when solving the system of equations
$y = 2x + 1$
and
$y=-x + 4$
using substitution, the solution is ______.
to verify the solution to a system of equations, you should ______ the solution back into the original equations.

Explanation:

First Question:

The method of solving a system of equations by plotting each equation on a coordinate plane (like a Cartesian plane) and identifying the point where they intersect (which represents the solution to the system) is known as the graphical method. This is a fundamental technique in algebra (a subfield of Mathematics) for solving systems of linear equations, where each equation is represented as a line, and their intersection gives the solution.

When dealing with real - world applications of systems of equations where the goal is to allocate hours (or time) to different activities in an optimal or organized way, this is related to time management. Time management often involves using mathematical models (like systems of equations) to distribute time resources among various tasks or activities.

Step1: Substitute \(y\) from first equation into second

We have the system \(

$$\begin{cases}y = 2x+1\\y=-x + 4\end{cases}$$

\). Since both expressions equal \(y\), we can set them equal to each other: \(2x + 1=-x+4\).

Step2: Solve for \(x\)

Add \(x\) to both sides: \(2x+x + 1=-x+x + 4\), which simplifies to \(3x+1 = 4\). Then subtract 1 from both sides: \(3x+1 - 1=4 - 1\), so \(3x=3\). Divide both sides by 3: \(x = 1\).

Step3: Find \(y\)

Substitute \(x = 1\) into the first equation \(y=2x + 1\). So \(y=2(1)+1=3\).

Answer:

graphical method

Second Question: