Question

solve the following system of linear equations by substitution and determine whether the system has one solution, no solution, or an infinite number of solutions. if the system has one solution, find the solution.\

$$\begin{cases} -4x + y = 12 \\\\ -10x + 2y = 26 \\end{cases}$$

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answer\
selecting an option will enable input for any required text boxes. if the selected option does not have any associated text boxes, then no further input is required.\
\bigcirc one solution \qquad (\boxed{\quad}, \boxed{\quad})\
\bigcirc no solution\
\bigcirc infinite number of solutions

Explanation:

Step1: Solve first equation for y

From \(-4x + y = 12\), we can express \(y\) in terms of \(x\) by adding \(4x\) to both sides. So we get \(y = 4x + 12\).

Step2: Substitute y into second equation

We substitute \(y = 4x + 12\) into the second equation \(-10x + 2y = 26\). This gives us \(-10x + 2(4x + 12)=26\).

Step3: Simplify and solve for x

First, expand the left - hand side: \(-10x+8x + 24 = 26\). Combine like terms: \(-2x+24 = 26\). Subtract 24 from both sides: \(-2x=26 - 24=2\). Then divide both sides by \(-2\): \(x=\frac{2}{-2}=-1\).

Step4: Find y using x

Now that we know \(x=-1\), we substitute it back into the equation \(y = 4x + 12\). So \(y=4\times(-1)+12=-4 + 12 = 8\).

Step5: Check the solution

Substitute \(x=-1\) and \(y = 8\) into the first equation: \(-4\times(-1)+8=4 + 8 = 12\), which matches the right - hand side of the first equation. Substitute into the second equation: \(-10\times(-1)+2\times8=10 + 16 = 26\), which matches the right - hand side of the second equation. So the system has one solution.

Answer:

One Solution \((-1, 8)\)