Question

solve each system by graphing.

1. $y = x + 1$

$y = 5x - 3$
solve each system by substitution.

1. $18x + 3y = 0$

$6x + y = 1$

1. $-x + y = 0$

$2x - 5y = 9$

Explanation:

Problem 30 (Solving by Graphing)

Step1: Analyze \( y = x + 1 \)

This is a linear equation in slope - intercept form (\(y=mx + b\)), where the slope \(m = 1\) and the y - intercept \(b=1\). To graph it, we can start by plotting the y - intercept at \((0,1)\). Then, using the slope (rise over run, since \(m = 1=\frac{1}{1}\)), we can find another point. From \((0,1)\), we move 1 unit up and 1 unit to the right to get the point \((1,2)\).

Step2: Analyze \( y=5x - 3 \)

This is also in slope - intercept form with slope \(m = 5\) and y - intercept \(b=-3\). Plot the y - intercept at \((0,-3)\). Using the slope \(m = 5=\frac{5}{1}\), from \((0,-3)\), we move 5 units up and 1 unit to the right to get the point \((1,2)\).

Step3: Find the Intersection

The two lines \(y=x + 1\) and \(y = 5x-3\) intersect at the point \((1,2)\) (we can see from the graphing steps that both lines pass through \((1,2)\)).

Step1: Solve for \(y\) from the second equation

We have the equation \(6x + y=1\). Solve for \(y\):
\(y=1 - 6x\)

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

The first equation is \(18x+3y = 0\). Substitute \(y = 1-6x\) into it:
\(18x+3(1 - 6x)=0\)
Expand the left - hand side: \(18x+3-18x = 0\)
Simplify: \(3=0\)
This is a contradiction, which means the system of equations \(

$$\begin{cases}18x + 3y=0\\6x + y=1\end{cases}$$

\) has no solution.

Step1: Solve for \(y\) from the first equation

We have the equation \(-x + y=0\). Solve for \(y\):
\(y=x\)

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

The second equation is \(2x-5y = 9\). Substitute \(y = x\) into it:
\(2x-5x=9\)
Combine like terms: \(-3x=9\)

Step3: Solve for \(x\)

Divide both sides of \(-3x = 9\) by \(-3\):
\(x=\frac{9}{-3}=-3\)

Step4: Solve for \(y\)

Since \(y=x\), then \(y=-3\)

Answer:

The solution of the system \(

$$\begin{cases}y=x + 1\\y=5x-3\end{cases}$$

\) is \(x = 1,y=2\) or the ordered pair \((1,2)\)

Problem 31 (Solving by Substitution)