Question

7 from unit 2, lesson 16 solve each system of equations without graphing. explain or show your reasoning. a. \\(\

$$\begin{cases} -5x + 3y = -8 \\\\ 3x - 7y = -3 \\end{cases}$$

\\) b. \\(\

$$\begin{cases} -8x - 2y = 24 \\\\ 5x - 3y = 2 \\end{cases}$$

\\)

Explanation:

Part a

Step1: Multiply equations to eliminate a variable

Let's use the elimination method. Multiply the first equation \(-5x + 3y = -8\) by \(3\) and the second equation \(3x - 7y = -3\) by \(5\) to make the coefficients of \(x\) opposite (in magnitude).
First equation after multiplying by \(3\): \(3\times(-5x + 3y)=3\times(-8)\) gives \(-15x + 9y = -24\)
Second equation after multiplying by \(5\): \(5\times(3x - 7y)=5\times(-3)\) gives \(15x - 35y = -15\)

Step2: Add the two new equations

Now add the two equations \(-15x + 9y = -24\) and \(15x - 35y = -15\) to eliminate \(x\):
\((-15x + 15x)+(9y - 35y)=-24 + (-15)\)
Simplify: \(0x - 26y = -39\) which is \(-26y = -39\)

Step3: Solve for \(y\)

Divide both sides of \(-26y = -39\) by \(-26\):
\(y=\frac{-39}{-26}=\frac{3}{2} = 1.5\)

Step4: Substitute \(y\) back to find \(x\)

Substitute \(y = \frac{3}{2}\) into the first original equation \(-5x + 3y = -8\):
\(-5x+3\times\frac{3}{2}=-8\)
\(-5x+\frac{9}{2}=-8\)
Subtract \(\frac{9}{2}\) from both sides: \(-5x=-8-\frac{9}{2}\)
\(-5x = \frac{-16 - 9}{2}=\frac{-25}{2}\)
Divide both sides by \(-5\): \(x=\frac{-25/2}{-5}=\frac{25}{10}=\frac{5}{2}=2.5\)

Step1: Simplify the first equation

Let's simplify the first equation \(-8x - 2y = 24\) by dividing all terms by \(-2\) to make the coefficients smaller.
\(\frac{-8x}{-2}-\frac{2y}{-2}=\frac{24}{-2}\) gives \(4x + y=-12\). Now we can express \(y\) in terms of \(x\): \(y=-12 - 4x\)

Step2: Substitute \(y\) into the second equation

Substitute \(y=-12 - 4x\) into the second equation \(5x - 3y = 2\):
\(5x-3(-12 - 4x)=2\)
Expand the left side: \(5x + 36 + 12x = 2\)

Step3: Combine like terms and solve for \(x\)

Combine \(5x\) and \(12x\): \(17x + 36 = 2\)
Subtract \(36\) from both sides: \(17x=2 - 36=-34\)
Divide both sides by \(17\): \(x=\frac{-34}{17}=-2\)

Step4: Substitute \(x\) back to find \(y\)

Substitute \(x = -2\) into \(y=-12 - 4x\):
\(y=-12-4\times(-2)=-12 + 8=-4\)

Answer:

\(x = \frac{5}{2}\), \(y=\frac{3}{2}\) (or \(x = 2.5\), \(y = 1.5\))

Part b