Question

u3 amy was given the following system of linear equations to solve. which of the following methods could amy do if she wanted to solve the system (select 2 answers)?

3x - 2y = 10

x + y = 4

□ solve by graphing both lines and find the x- and y- intercepts

□ solve by graphing both lines and find where they intersect

✓ solve by creating a 2x2 and 2x1 matrix to solve

□ solve by square rooting both equations to get y by itself

Explanation:

Step1: Analyze each method

• Solve by graphing intercepts: Graphing intercepts and finding where lines meet (intersection) is valid for linear systems. But the option says "find x - and y - intercepts" (not about intersection), but the other graphing option is about intersection. Wait, the first graphing option: solving by graphing lines and finding intercepts (but to solve a system, we need intersection). The second graphing option: "find where they intersect" – that's correct for solving a system (the solution is the intersection point).
• Solve by graphing intersection: A system of linear equations' solution is the intersection point of the two lines. So graphing both lines and finding their intersection is a valid method.
• Matrix method: For a system \(

$$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$

\), we can represent it as a \( 2\times2 \) matrix \(

$$\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}$$

\) (coefficient matrix) and a \( 2\times1 \) matrix \(

$$\begin{bmatrix} c_1 \\ c_2 \end{bmatrix}$$

\) (constant matrix), then use matrix methods (like Cramer's rule, inverse matrix) to solve. So this is valid.

• Square rooting: The equations are linear (\( x \) and \( y \) are to the first power), square rooting is for equations with squares (like \( x^2 + y^2 = r^2 \)), so this method is invalid.

Now, the correct methods are: "Solve by graphing both lines and find where they intersect" and "Solve by creating a 2x2 and 2x1 matrix to solve". Wait, the question says "SELECT 2 answers". Let's re - check:

1. "Solve by graphing both lines and find the x - and y - intercepts": This is about finding intercepts of each line, not the solution of the system (which is intersection). So this is incorrect.
2. "Solve by graphing both lines and find where they intersect": Correct, as the solution of the system is the intersection point.
3. "Solve by creating a 2x2 and 2x1 matrix to solve": Correct, as linear systems can be solved using matrix methods (e.g., representing as \( Ax = b \), where \( A \) is \( 2\times2 \), \( x \) is \( 2\times1 \), \( b \) is \( 2\times1 \)).
4. "Solve by square rooting both equations": The equations are linear, no squares, so this is incorrect.

So the two correct options are: "Solve by graphing both lines and find where they intersect" and "Solve by creating a 2x2 and 2x1 matrix to solve". Wait, but in the original, one is already checked (matrix), the other correct is the graphing intersection.

Step2: Confirm the correct options

• For the graphing method: The solution to a system of two linear equations is the point where the two lines intersect. So graphing both lines and finding their intersection is a valid way to solve the system.
• For the matrix method: The given system \(

$$\begin{cases} 3x - 2y = 10 \\ x + y = 4 \end{cases}$$

\) can be written as \(

$$\begin{bmatrix} 3 & - 2 \\ 1 & 1 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \end{bmatrix}$$

=

$$\begin{bmatrix} 10 \\ 4 \end{bmatrix}$$

\), so using a \( 2\times2 \) (coefficient) and \( 2\times1 \) (constant) matrix to solve is valid.

• The first graphing option is about intercepts (not solving the system), and the square - rooting option is invalid for linear equations.

Answer:

B. Solve by graphing both lines and find where they intersect
C. Solve by creating a 2x2 and 2x1 matrix to solve

(Assuming the options are labeled as:
A. Solve by graphing both lines and find the x - and y - intercepts
B. Solve by graphing both lines and find where they intersect
C. Solve by creating a 2x2 and 2x1 matrix to solve
D. Solve by square rooting both equations to get y by itself)