Solve the system of linear equations by graphing. y = -4x + 3 4x - 2y = 6

Understand the Problem

The question is asking us to solve a system of linear equations by graphing. We need to graph both equations and find the point where they intersect.

Answer

The solution to the system of equations is the point $ (1, -1) $.

Steps to Solve

Graph the first equation

The first equation is already in slope-intercept form $y = mx + b$, where $m = -4$ and $b = 3$.

Start by plotting the y-intercept at $(0, 3)$.
Use the slope to find another point. The slope $-4$ means for every 1 unit you move right ($+1$ in $x$), you move down 4 units in $y$. From $(0, 3)$, the next point is $(1, -1)$.
Draw a line through these points.

Rearrange the second equation

The second equation is $4x - 2y = 6$.

To make it easier to graph, solve for $y$: $$ 4x - 2y = 6 $$ $$ -2y = -4x + 6 $$ $$ y = 2x - 3 $$

Graph the second equation

Now that we have $y = 2x - 3$, which is in slope-intercept form where $m = 2$ and $b = -3$.

Plot the y-intercept at $(0, -3)$.
Use the slope to find another point. The slope $2$ means for every 1 unit you move right ($+1$ in $x$), you move up 2 units in $y$. From $(0, -3)$, the next point is $(1, -1)$.
Draw a line through these points.

Find the intersection point

Now, look for the point where the two lines intersect. The lines intersect at the point $(1, -1)$.

The solution to the system of equations is the point of intersection: $ (1, -1) $.

More Information

This point means that when $x = 1$, both equations yield $y = -1$. Thus, $(1, -1)$ is the solution to the system. Graphing is a visual method to find solutions to systems of equations.

Tips

Not plotting points accurately can lead to an incorrect intersection.
Forgetting to correctly convert the second equation to slope-intercept form makes graphing difficult. Always double-check your algebra when rearranging equations.

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