Question

which graph represents the system of equations? $y = \frac{2}{5} x +1$
$y = -\frac{1}{5} x+1$
a. b. c. d.
\bigcirc a.
\bigcirc b.
\bigcirc c
\bigcirc d.

1. solve the system $x = y + 4$ and $x + y = 10$ enter your answer as an ordered pair

your answer

Explanation:

Question 6 (Graph of System of Equations)

Step1: Analyze the equations' form

Both equations are in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For \(y=\frac{2}{5}x + 1\) and \(y=-\frac{1}{5}x + 1\), the y - intercept \(b = 1\) for both. So both lines should cross the y - axis at \((0,1)\).

Step2: Analyze the slopes

The slope of the first line \(m_1=\frac{2}{5}\) (positive, so the line rises from left to right) and the slope of the second line \(m_2 =-\frac{1}{5}\) (negative, so the line falls from left to right).
Now let's analyze the options:

• Option A: The lines do not seem to have the same y - intercept. Eliminate A.
• Option B: The lines pass through the origin \((0,0)\), not \((0,1)\). Eliminate B.
• Option C: Both lines have a y - intercept of 1 (cross the y - axis at \((0,1)\)), one line has a positive slope (rising) and one has a negative slope (falling), which matches our analysis.
• Option D: The lines do not have the same y - intercept. Eliminate D.

Step1: Substitute \(x\) into the second equation

We have the system \(

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

\). Substitute \(x=y + 4\) from the first equation into the second equation: \((y + 4)+y=10\).

Step2: Solve for \(y\)

Simplify the equation: \(y+4 + y=10\), combine like terms: \(2y+4 = 10\). Subtract 4 from both sides: \(2y=10 - 4=6\). Divide both sides by 2: \(y=\frac{6}{2}=3\).

Step3: Solve for \(x\)

Substitute \(y = 3\) into the first equation \(x=y + 4\), so \(x=3 + 4=7\).

Answer:

C

Question 7 (Solving the System of Equations)