Find the Points of Intersection y=2x+1 , y=-1/2x+21

$y = 2 x + 1$ , $y = - \frac{1}{2} x + 21$

Step 1

Eliminate the equal sides of each equation and combine.

$2 x + 1 = - \frac{1}{2} x + 21$

Step 2

Solve $2 x + 1 = - \frac{1}{2} x + 21$ for $x$ .

Combine $x$ and $\frac{1}{2}$ .

$2 x + 1 = - \frac{x}{2} + 21$

Move all terms containing $x$ to the left side of the equation.

Add $\frac{x}{2}$ to both sides of the equation.

$2 x + 1 + \frac{x}{2} = 21$

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x \cdot \frac{2}{2} + \frac{x}{2} + 1 = 21$

Combine $2 x$ and $\frac{2}{2}$ .

$\frac{2 x \cdot 2}{2} + \frac{x}{2} + 1 = 21$

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 + x}{2} + 1 = 21$

Simplify each term.

Simplify the numerator.

Factor $x$ out of $2 x \cdot 2 + x$ .

Factor $x$ out of $2 x \cdot 2$ .

$\frac{x (2 \cdot 2) + x}{2} + 1 = 21$

Raise $x$ to the power of $1$ .

$\frac{x (2 \cdot 2) + x^{1}}{2} + 1 = 21$

Factor $x$ out of $x^{1}$ .

$\frac{x (2 \cdot 2) + x \cdot 1}{2} + 1 = 21$

Factor $x$ out of $x (2 \cdot 2) + x \cdot 1$ .

$\frac{x (2 \cdot 2 + 1)}{2} + 1 = 21$

Multiply $2$ by $2$ .

$\frac{x (4 + 1)}{2} + 1 = 21$

Add $4$ and $1$ .

$\frac{x \cdot 5}{2} + 1 = 21$

Move $5$ to the left of $x$ .

$\frac{5 x}{2} + 1 = 21$

Move all terms not containing $x$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$\frac{5 x}{2} = 21 - 1$

Subtract $1$ from $21$ .

$\frac{5 x}{2} = 20$

Multiply both sides of the equation by $\frac{2}{5}$ .

$\frac{2}{5} \cdot \frac{5 x}{2} = \frac{2}{5} \cdot 20$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{2}{5} \cdot \frac{5 x}{2}$ .

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2}{5} \cdot \frac{5 x}{2} = \frac{2}{5} \cdot 20$

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{2}{5} \cdot 20$

Cancel the common factor of $5$ .

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{2}{5} \cdot 20$

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{2}{5} \cdot 20$

Rewrite the expression.

$x = \frac{2}{5} \cdot 20$

Simplify the right side.

Simplify $\frac{2}{5} \cdot 20$ .

Cancel the common factor of $5$ .

Factor $5$ out of $20$ .

$x = \frac{2}{5} \cdot (5 (4))$

Cancel the common factor.

$x = \frac{2}{5} \cdot (5 \cdot 4)$

Rewrite the expression.

$x = 2 \cdot 4$

Multiply $2$ by $4$ .

$x = 8$

Step 3

Evaluate $y$ when $x = 8$ .

Substitute $8$ for $x$ .

$y = - \frac{1}{2} \cdot 8 + 21$

Substitute $8$ for $x$ in $y = - \frac{1}{2} \cdot 8 + 21$ and solve for $y$ .

Remove parentheses.

$y = - \frac{1}{2} \cdot 8 + 21$

Simplify $- \frac{1}{2} \cdot 8 + 21$ .

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y = \frac{- 1}{2} \cdot 8 + 21$

Factor $2$ out of $8$ .

$y = \frac{- 1}{2} \cdot (2 (4)) + 21$

Cancel the common factor.

$y = \frac{- 1}{2} \cdot (2 \cdot 4) + 21$

Rewrite the expression.

$y = - 1 \cdot 4 + 21$

Multiply $- 1$ by $4$ .

$y = - 4 + 21$

Add $- 4$ and $21$ .

$y = 17$

Step 4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(8, 17)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(8, 17)$

Equation Form:

$x = 8, y = 17$

Step 6

Find the Points of Intersection y=2x+1 , y=-1/2x+21

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