Use the four-step strategy to solve each problem. Use x, y,...

Use the four-step strategy to solve each problem. Use $x, y,$ and $z$ to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. The bar graph shows the average annual spending per person on selected items in 1980 and $2010 .$ All dollar amounts are adjusted for inflation. Annual Spending per Person in the United States, Adjusted for Inflation CAN'T COPY THE FIGURE In this exercise, we refer to annual spending per person in $2010 .$ The combined spending on housing, vehicles/gas, and health care was $\$ 13,840 .$ The difference between spending on housing and spending on vehicles/gas was \$3864. The difference between spending on housing and spending on health care was $\$ 695 .$ Find the average per-person spending on housing, vehicles/gas, and health care in 2010 .

Use the four-step strategy to solve each problem. Use $x, y,$ and $z$ to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables.
The bar graph shows the average annual spending per person on selected items in 1980 and $2010 .$ All dollar amounts are adjusted for inflation.
Annual Spending per Person in the United States, Adjusted for Inflation
CAN'T COPY THE FIGURE
In this exercise, we refer to annual spending per person in $2010 .$ The combined spending on housing, vehicles/gas, and health care was $\$ 13,840 .$ The difference between spending on housing and spending on vehicles/gas was \$3864. The difference between spending on housing and spending on health care was $\$ 695 .$ Find the average per-person spending on housing, vehicles/gas, and health care in 2010 .

Key Concepts

Solving Techniques for Systems

Once the system has been established, techniques such as substitution, elimination, or using matrices can be employed to solve for the unknown values. These methods help systematically reduce the complexity of the system until the values of the variables are found.

System of Linear Equations

A system of linear equations is a collection of equations where all the equations are linear and share the same set of unknowns. The goal is to find a unique set of values for these unknowns that simultaneously satisfies all the equations.

Variables Representing Unknown Quantities

Using symbols like x, y, and z to represent unknown quantities allows us to abstractly refer to values that need to be determined. This is crucial in translating real-world problems into algebraic form.

Translating Word Problems into Equations

This process involves reading the verbal descriptions provided in a problem, identifying the relationships between quantities, and then expressing those relationships as algebraic equations. This step converts real-world information into a solvable mathematical format.

Transcript

00:01 For this problem, we have a set of statements about how the spending on housing, vehicles, and health care, and we want to solve for the exact amount of spending on each one.

00:14 And so here i've written out that when i say x, that refers to housing, y refers to vehicles, and z refers to health care.

00:23 And so our first statement is that the combined spending on housing, vehicles, and health care was 13 ,8 ,000.

00:31 And $40.

00:32 And so we can write that out as our first equation as the combined spending on all three.

00:38 So all of them added together, x plus y plus z, is equal to 13, 840.

00:47 And that's in dollars.

00:49 And so our second statement then is the difference between the spending on housing and the spending on vehicles was 3864.

00:59 So the spending on housing, which is x, minus the spending on vehicles is 3864.

01:08 And then finally, the difference on spending between housing and healthcare was $695.

01:16 So housing minus healthcare is $695.

01:21 So now we have a set of three equations that we can solve by elimination in this case.

01:28 And so first of all, we can see that these first two equations, have a coefficient of y as 1.

01:37 And so we can add the first two equations together to cancel out the y's.

01:44 So adding these two equations, 1 plus 2, the x values, we have x plus x, which gives us 2x.

01:54 For the y values, we have y minus y, which gives us 0, and then z, which is just c...

Please give Ace some feedback