Answered

Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

On your own paper, graph the system of equations and identify the solution.

[tex]\[
\begin{array}{l}
y = 2x - 4 \\
y = -\frac{1}{2}x + 1
\end{array}
\][/tex]

A. [tex]\((3, 2)\)[/tex]
B. [tex]\((0, 2)\)[/tex]
C. [tex]\((2, 0)\)[/tex]
D. [tex]\((2, 3)\)[/tex]

Sagot :

GinnyAnswer
To find the solution to the system of equations [tex]\[ \begin{array}{l} y = 2x - 4 \\ y = -\frac{1}{2}x + 1 \end{array} \][/tex]
we need to find the point where these two lines intersect. Here is a detailed, step-by-step explanation of how to solve this system of equations and identify the solution:

### Step 1: Set the Equations Equal
Since both equations are equal to [tex]\(y\)[/tex], we can set the right-hand sides equal to each other:
[tex]\[2x - 4 = -\frac{1}{2}x + 1\][/tex]

### Step 2: Combine Like Terms
To solve for [tex]\(x\)[/tex], we first need to combine like terms. Add [tex]\(\frac{1}{2}x\)[/tex] to both sides:
[tex]\[2x + \frac{1}{2}x - 4 = 1\][/tex]

### Step 3: Simplify the Equation
Combine the [tex]\(x\)[/tex] terms on the left-hand side. Note that [tex]\(2x\)[/tex] is equivalent to [tex]\(\frac{4}{2}x\)[/tex], so:
[tex]\[\frac{4}{2}x + \frac{1}{2}x = 5\][/tex]

Adding these fractions, we get:
[tex]\[\frac{5}{2}x - 4 = 1\][/tex]

### Step 4: Solve for [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], add 4 to both sides of the equation:
[tex]\[\frac{5}{2}x = 5\][/tex]

Next, multiply both sides by [tex]\(\frac{2}{5}\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[x = 2\][/tex]

### Step 5: Substitute [tex]\(x\)[/tex] Back into One of the Original Equations
Now that we have [tex]\(x = 2\)[/tex], we substitute this value back into one of the original equations to find [tex]\(y\)[/tex]. Using the first equation:
[tex]\[y = 2x - 4\][/tex]
[tex]\[y = 2(2) - 4\][/tex]
[tex]\[y = 4 - 4\][/tex]
[tex]\[y = 0\][/tex]

### Solution
The solution to the system of equations is the point [tex]\((2, 0)\)[/tex].

### Verification
To verify, substitute [tex]\(x = 2\)[/tex] into the second equation:
[tex]\[y = -\frac{1}{2}(2) + 1\][/tex]
[tex]\[y = -1 + 1\][/tex]
[tex]\[y = 0\][/tex]

### Graphical Solution
When plotted on a graph, the lines represented by [tex]\(y = 2x - 4\)[/tex] and [tex]\(y = -\frac{1}{2}x + 1\)[/tex] will intersect at the point [tex]\((2, 0)\)[/tex].

### Conclusion
Thus, the solution to the system of equations is:

[tex]\((2, 0)\)[/tex].

So, the correct answer from the given options is:

[tex]\[ \boxed{(2, 0)} \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.