Problem 38 - Elementary Algebra

Question

Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x+y=4 \\ x=1 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( (1, 3) \).

Step 1 - Graph the first equation

Rewrite the first equation in slope-intercept form: \[ x + y = 4 \] Subtract \( x \) from both sides: \[ y = -x + 4 \] Now, plot this line on the graph. First, find the y-intercept (when \( x = 0 \)): \[ y = 4 \] This gives the point \( (0, 4) \).Next, use the slope to find a second point. The slope is \(-1\) (or \(-1/1\)), so from \( (0, 4) \), go down 1 unit and right 1 unit which gives the point \( (1, 3) \). Draw the line through these points.

Step 2 - Graph the second equation

The second equation is \( x = 1 \). This is a vertical line that crosses the x-axis at \( x = 1 \). Simply draw a vertical line passing through \( x = 1 \).

Step 3 - Find the intersection point

Look at the graph and find the point where the two lines intersect. The line \( y = -x + 4 \) intersects the vertical line \( x = 1 \) at the point where \( x = 1 \). Substitute \( x = 1 \) into the first equation to find \( y \): \[ 1 + y = 4 \] Subtract 1 from both sides: \[ y = 3 \] So the intersection point is \( (1, 3) \).

Key Concepts

Graphing Linear Equations

Graphing linear equations is a key skill in solving systems of equations visually. Linear equations can be represented on a graph as straight lines. To start graphing a linear equation, it's helpful to rewrite it in the slope-intercept form, which makes it easier to plot. Here's a quick breakdown:

- Start by transforming the equation into the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Identify the y-intercept (\